1896 


LIBRARY 


UNIVERSITY  OF  CALIFORNIA. 


01  FT  OR 


Received 
Accession  No. 


Class  No. 


THE  ELEMENTS 


OF   THE 


FOUB  INNER  PLANETS 


AND   THE 


FUNDAMENTAL  CONSTANTS  OF  ASTRONOMY 


BY 


SIMON    NEWCOMB 


Supplement  to  the  American  Ephemeris  and  Nautical 
Almanac  for  1897 


WASHINGTON 

GOVERNMENT  PRINTING  OFFICE 
1895 


PREFACE. 


THE  diversity  in  the  adopted  values  of  the  elements  and 
constants  of  astronomy  is  productive  of  inconvenience  to  all 
who  are  engaged  in  investigations  based  upon  these  quanti- 
ties, and  injurious  to  the  precision  and  symmetry  of  much  of 
our  astronomical  work.  If  any  cases  exist  in  which  uniform 
and  consistent  values  of  all  these  quantities  are  embodied  in 
an  extended  series  of  astronomical  results,  whether  in  the 
form  of  ephemerides  or  results  of  observations,  they  are  the 
exception  rather  than  the  rule.  The  longer  this  diversity 
continues  the  greater  the  difficulties  which  astronomers  of 
the  future  will  meet  in  utilizing  the  work  of  our  time. 

On  taking  charge  of  the  work  of  preparing  the  American 
Ephemeris  in  1877  the  writer  was  so  strongly  impressed  with 
the  inconvenience  arising  from  this  source  that  he  deemed  it 
advisable  to  devote  all  the  force  which  he  could  spare  to  the 
work  of  deriving  improved  values  of  the  fundamental  elements 
and  embodying  them  in  new  tables  of  the  celestial  motions. 
It  was  expected  that  the  work  could  all  be  done  in  ten  years. 
But  a  number  of  circumstances,  not  necessary  to  describe  at 
present,  prevented  the  fulfillment  of  this  hope.  Only  now  is 
the  work  complete  so  far  as  regards  the  fundamental  constants 
and  the  elements  of  the  planets  from  Mercury  to  Jupiter  inclu- 
sive. The  construction  of  tables  of  the  four  inner  planets  is 
now  in  progress,  those  of  Jupiter  and  Saturn  having  already 
been  completed  by  Mr.  HILL.  All  these  tables  will  be  pub- 
lished as  soon  as  possible,  and  the  investigations  on  which 
they  are  based  are  intended,  so  far  as  it  is  practicable  to  con- 
dense them,  to  appear  in  subsequent  volumes  of  the  Astro- 
nomical Papers  of  the  American  Ephemeris.  As  it  will  take 
several  years  to  bring  out  these  volumes,  it  has  been  deemed 
advisable  to  publish  in  advance  the  present  brief  summary  of 

the  work. 

HI 


IV  PREFACE. 

The  author  feels  that  critical  examination  of  this  monograph 
may  show  in  many  points  a  want  of  consistency  and  conti- 
nuity. The  ground  covered  is  so  extensive,  the  material  so 
diverse  as  well  as  voluminous,  and  the  relations  to  be  investi- 
gated so  numerous,  that  no  conclusion  could  be  reached  on 
one  point  which  was  not  liable  to  be  modified  by  subsequent 
decisions  upon  other  points.  The  author  trusts  that  the  diffi- 
culties growing  out  of  these  features  of  the  work,  as  well  as 
those  incident  to  the  administration  of  an  office  not  especially 
organized  for  the  work,  will  afford  a  sufficient  apology  for  any 
defects  that  may  be  noticed. 

NAUTICAL  ALMANAC  OFFICE, 

U.  8.  Naval  Observatory,  January  7, 1895. 


I^IVWITT 

CONTENTS. 


CHAPTER  I.— GENERAL  OUTLINE  OF  THE  WORK  OF  COMPARING 

THE   OBSERVATIONS   WITH   THEORY. 

Page. 
§    1.  Reduction  to  the  standard  system  of  Right  Ascensions  and 

Declinations 

§    2.  Observations  used 

§    3.  Semidiameters  of  Mercury  and  Venus. — Table  for  defective 

illumination  of  Mercury  in  Right  Ascension 3 

§    4.  Tabular  places  from  LEVERRIER'S  tables. — Reduction  for 

masses  used  by  LEVERRIER 6 

§    5.  Comparisons  of  observations  and  tables 8 

$    6.  Equations  of  condition. — Method  of  formation 8 

§    7.  Method  of  determining  the  secular  variations  and  the  masses 

of  Venus  and  Mercury  independently 10 

§  8.  Method  of  introducing  the  results  of  observations  on  transits 
of  Venus  and  Mercury ;  separate  solutions,  A  from  meridian 
observations  without  transits ;  B,  including  both  meridian 
observations  and  transits 13 

CHAPTER  II.— DISCUSSION  AND  RESULTS  OF  OBSERVATIONS  OF 

THE  SUN. 

$  9.  Method  of  treating  observed  Right  Ascensions  of  the  Sun.— 
Expression  of  errors  of  observed  Right  Ascension  as  error 
of  longitude 15 

§  10.  Treatment  of  observed  Declinations  of  the  Sun. — Formation 
of  equations  of  condition  for  the  corrections  to  the 
obliquity  and  to  the  Sun's  absolute  longitude 16 

$  11.  Formation  of  equations  from  observed  Right  Ascensions  of 

Sun 17 

§  12.  Solution  of  equations  from  Right  Ascensions  of  the  Sun. — 
Tabular  exhibit  of  results  of  observations  of  the  Sun's 
Right  Ascensions  at  various  observatories  during  different 
periods 20 

§  13.  Mass  of  Venus,  derived  from  observations  of  the  Sun's  Right 

Ascension 24 

$  14.  Discussion  of  corrections  to  the  Right  Ascensions  of  the  Sun 

relative  to  that  of  the  stars 25 

v 


VI  CONTENTS. 

Page. 

§  15.  Discussion  of  corrections  to  the  eccentricity  and  perihelion 

of  the  Earth's  orbit 27 

§  16.  Results  of  observed  Declinations  of  the  Sun.— Exhibit  of 
individual  corrections  to  the  absolute  longitude  and  the 
obliquity  of  the  ecliptic  at  the  different  observatories 
during  different  periods ^ 29 

$  17.  Discussion  of  the  observed  corrections  to  the  Sun's  absolute 

longitude 32 

§  18.  Discussion  of  the  observed  corrections  to  the  obliquity  of 

the  ecliptic 33 

$  19.  Effect  of  refraction  on  the  obliquity ;  special  investigation  of 
the  secular  change  of  obliquity  as  derived  from  observa- 
tions of  the  Sun 35 

$  20.  Concluded  results  for  the  obliquity,  and  its  secular  varia- 
tion    39 

§  21.  Summary  of  results  for  the  corrections  to  the  elements  of  the 
Earth's  orbit  and  their  secular  variations  as  derived  from 
observations  of  the  Sun  alone . 41 

CHAPTER    III. — RESULTS    OF    OBSERVATIONS    OF    THE    PLANETS 
MERCURY,  VENUS,  AND  MARS. 

§  22.  Elements  adopted  for  correction 43 

$  23.  Introduction  .of  the  corrections  to  the  masses  of  Venus  and 

Mercury 45 

§  24.  Introduction  of  the  errors  of  absolute  Right  Ascension  and 

Declinations  of  the  standard  stars 46 

§  25.  Introduction  of  the  corrections  to  the  secular  variations. — 
Method  of  forming  the  normal  equations  by  periods  so  as 

to  include  the  correction  to  the  secular  variation 49 

$  26.  Dates  and  weights  of  the  equations  for  the  various  periods.  52 

§  27.  Unknown  quantities  of  the  equations. — Factors  for  changing 
corrections  of  the  unknown  quantities  into  corrections  of 

the  elements 55 

§  28.  Table  of  the  values  of  the  principal  coefficients  of  the  normal 

equations '..'...' 56 

§  29.  Order  of  elimination 57 

§  30.  Treatment  of  meridian  observations  of  Mercury. — Effect  of 
want  of  approximation  in  the  coefficients  of  the  equations 

of  condition 58 

§  31.  Introduction  of  the  equations  derived  from  observed  tran- 
sits of  Mercury 61 

$  32.  Solution  of  the  equations  for  Mercury 65 

$  33.  Systematic  discordances  among  the  observed  Right  Ascen- 
sions of  Mercury  in  different  points  of  its  relative  orbit..  66 


CONTENTS.  VII 

Page. 

34.  Comparison  of  the  results  derived  from  meridian  observa- 

tions of  Mercury  with  those  derived  from  transits  over  the 

Sun'sdisk  69 

35.  Treatment  of  meridian  observations  of  Venus 70 

36.  Results  of  observed  transits  of  Venus 70 

37.  Equations  derived  from  observed  transits  of  Venus 75 

38.  Solutions  of  the  equations  from  Venus 76 

39.  Comparison  of  the  results  of  meridian  observations  of  Venus 

with  those  of  transits 76 

40.  Solution  of  the  equations  for   Mars.— Inequality   of   long 

period  in  the  mean  longitude  and  perihelion,  indicated  by. 

observations 77 

41.  Reduction  from  the  equator  to  the  ecliptic 79 


CHAPTER  IV. — COMBINATION    OF    THE    PRECEDING    RESULTS   TO 

OBTAIN  THE  MOST  PROBABLE  VALUES  OF  THE  ELEMENTS 
AND  OF  THEIR  SECULAR  VARIATIONS  FROM  OBSERVA- 
TIONS ALONE. 

§  42.  Modifications  of  the  canons  of  least  squares 81 

§  43.  Relative  precision  of  the  two  methods  of  determining  the 

elements  of  the  Earth's  orbit 86 

$  44.  Concluded    secular    variations   of   the    solar   elements,    as 

derived  from  observations  alone 87 

$  45.  Common  error  of  the  standard  declinations 89 

$  46.  Definitive  secular  variations  of  all  the  elements  from  obser- 
vations alone. — Matrices  of  the  normal  equations  for  the 

secular  variations. — Tabular  statement  of  results 90 

$  47.  Definitive  corrections  to  the  solar  elements  for  1850. .  95 


CHAPTER  V. — MASSES  OF   THE  PLANETS  DERIVED   BY  METHODS 

INDEPENDENT    OF     THE     SECULAR    VARIATIONS,    WITH   THE 
RESULTING   COMPUTED    SECULAR   VARIATIONS. 

$  48.  Plan  of  discussion 97 

§  49.  Mass  of  Jupiter ;  general  combination  of  results 97 

§  50.  Mass  of  Mars. — Prof.  HALL'S  value  adopted 99 

§  51.  Mass  of  the  Earth,  derived  from  the  preliminary  value  of  the 

solar  parallax 99 

§  52.  Mass  of  Venus,  derived  from  periodic  perturbations 101 

§  53.  Mass  of  Mercury,  from  various  sources 102 

§  54.  Theoretical  values  of  the  secular  variations  for  1850...  106 


VIII  CONTENTS. 

Page. 

CHAPTER  VI. — EXAMINATION  OF  HYPOTHESES  AND  DETERMINA- 
TION OF  THE  MASSES  BY  WHICH  THE  DEVIATIONS  OF  THE 
SECULAR  VARIATIONS  FROM  THEIR  THEORETICAL  VALUES 
MAY  BE  EXPLAINED. 

§  55.  Comparison  of  the  observed  and  theoretical  secular  varia- 
tions    109 

$  56.  Hypothesis  of  nonsphericity  of  the  equipotential  surfaces 

of  the  Sun Ill 

§  57.  Hypothesis  of  an  intraniercurial  ring 112 

§  58.  Hypothesis  of  an  extended  mass  of  diffused  matter,  like  that 

Which  reflects  the  zodiacal  light 115 

$  59.  Hypothesis  of  a  ring  of  planets  outside  the  orbit  of  Mer- 
cury.— Elements  of  such  a  ring. — This  hypothesis  the  only 
one  which  represents  the  observations,  but  too  improbable 
to  be  accepted 116 

$  60.  Examination  of  the  question  whether  the  excess  of  motion 
of  the  perihelion  of  Mars  may  be  due  to  the  action  of  the 
zone  of  minor  planets 116 

§  61.  Hypothesis  that  gravitation  toward  the  Sun  is  not  exactly 

as  the  inverse  square  of  the  distance 118 

§  62.  Degree  of  precision  with  which  the  theory' of  the  inverse 

square  is  established 119 

§  63.  Determination  of  the  masses  which  will  best  represent  the 
observed  secular  variations  of  the  eccentricities,  nodes, 
and  inclinations 121 

$  64.  Preliminary  adjustment  of  the  two  sets  of1  masses. — Result- 
ing value  of  the  solar  parallax 122 

CHAPTER  VII. — VALUES   OF  THE  PRINCIPAL  CONSTANTS  WHICH 

DEPEND   UPON   THE   MOTION   OF   THE   EARTH. 

§  65.  The  processional  constant 124 

§  66.  The  constant  of  nutation,  derived  from  observations 129 

§  67.  Relations  between  the  constants  of  precession  and  nutation 

and  the  quantities  on  which  they  depend 131 

$  68.  The  mass  of  the  Moon  from  the  observed  constant  of  nuta- 
tion   ---  132 

§  69.  The  constant  of  aberration 133 

§  70.  The  values  of  this  constant,  derived  from  observations 135 

§  71.  The  lunar  inequality  in  the  Earth's  motion 139 

§  72.  The  solar  parallax  derived  from  the  lunar  inequality 142 

$  73.  Values  of  the  solar  parallax  derived  from  measurements  of 
Venus  on  the  face  of  the  Sun  during  the  transits  of  1874 

and  1882,  with  the  heliometer  and  photoheliograph 143 

$  74.  The  solar  parallax  from  observed  contacts  during  transits  of 

Venus.. 145 


CONTENTS.  IX 


§  75.  Solar  parallax  from  the  observed  constant  of  aberration  and 

measured  velocity  of  light 147 

§    76.  Solar  parallax  from  the  parallactic  inequality  of  the  Moon.  148 

§i  77.  Solar  parallax  from  observations  of  the  minor  planets  with 

the  heliometer 152 

§  78.  Remarks  on  determinations  of  the  parallax  which  are  not 
used  in  the  present  discussion. — Errors  arising  from  dif- 
ferences of  color 154 

CHAPTER  VIII. — DISCUSSION  OF  RESULTS  FOR  THE  SOLAR  PARAL- 
LAX  AND   THE   MASSES   OF   THE   THREE   INNER   PLANETS. 

§    79.  Separate  values  of  the  solar  parallax,   and  their  general 

mean 156 

§    80.  Rediscussion  of  the  motion  of  the  node  of  Venus 159 

§    81.  Possible  systematic  errors  in  determinations  of  the  parallax.  164 

§    82.  Revised  list  of  determinations 166 

§    83.  Definitive   adjustment  of  the   masses  of  the   three  inner 

planets 168 

§    84.  Possible  causes  of  the  observed  discordances 173 

§    85.  Adopted  values  of  the  doubtful  quantities 173 

§    86.  Bearing  of  future  determinations  on  the  question. 175 

CHAPTER  IX. — DERIVATION  OF  RESULTS. 

§    87.  Ulterior  corrections  to  the  motions  of  the  perihelion  and 

mean  longitude  of  Mercury 178 

§    88.  Definitive  elements  of  the  four  inner  planets  for  the  epoch 

1850,  as  inferred  from  all  the  data  of  observation 179 

$    89.  Definitive  values  of  the  secular  variations 182 

§    90.  Secular  acceleration  of  the  mean  motions 186 

§    91.  The  measure  of  time 188 

§    92.  The  constant  of  aberration 188 

§    93.  The  mass  of  the  Moon 189 

§    94.  The  parallactic  inequality  of  the  Moon 190 

§    95.  The  centimeter-second  system  of  astronomical  units 190 

\S    96.  Masses  of  the  Earth  and  Moon  in  centimeter-second  units..  191 

^S    97.  Parallax  of  the  Moon 193 

§    98.  Mass  and  parallax  of  the  Sun 194 

§    99.  Constant  of  nutation,   and   mechanical  ellipticity  of  the 

Earth 195 

§  100.  Precession  196 

$  101 .  Obliquity  of  the  ecliptic 196 

§  102.  Relative  positions  of  the  equator  and  the  ecliptic  at  differ- 
ent epochs  for  reduction  of  places  of  stars  and  planets  . .  197 


ELEMENTTlfFCONSTANTS. 


OHAPTEK  1. 

GENERAL  OUTLINE    OF  THE  WORK    OF  COMPARING  THE 
OBSERVATIONS  WITH  THEORY. 

1.  In  logical  order,  the  first  step  in  the  work  consists  in  the 
reduction  of  observed  positions  of  the  Sun  and  planets  to  a 
uniform  equinox  and  system  of  declinations. 

The  adopted  standard  of  Eight  Ascensions  was  that  origi- 
nally worked  out  in  my  paper  on  the  Eight  Ascensions  of  the 
fundamental  stars,  found  in  an  appendix  to  the  Washington 
Observations  for  1870,  and  extended  to  a  fundamental  system 
of  time  stars  in  the  catalogue  published  in  Yol.  1  of  the  Astro- 
nomical Papers  of  the  American  Ephemeris.  This  system 
coincides  closely  with  that  of  the  Astronomische  Gesellschaft 
and  the  Berliner  Jahrbuch,  about  the  epoch  1870,  but  the  cen- 
tennial proper  motion  is  greater'  by  about  08.08. 

In  Declinations,  the  adopted  standard  was  that  of  Boss, 
which  has  been  used  in  the  American  Ephemeris  since  1881, 
and  on  which  is  based  the  catalogue  of  zodiacal  stars  just 
referred  to.  But  as  Declinations  generally  are  not  immediately 
referred  to  fundamental  stars,  the  method  of  reducing  obser- 
vations to  this  system  in  Declination  was  not  entirely  uniform. 

Observations  used. 

2.  The  following  is  a  general  statement  of  the  observations 
used,  and  the  extent  to  which  they  were  corrected,  or  re-re- 
duced. 

Greenwich. — Dr.  AUWERS  courteously  supplied  me  with  the 
results  of  his  re-reduction  of  BRADLEY'S  observations  both  of 
the  Sun  and  planets.  From  the  beginning  of  MASKYLENE'S 
work  until  1835,  the  Greenwich  observations  were  completely 
re-reduced,  utilizing,  so  far  as  possible,  AIRY'S  reductions.  The 
5690  N  ALM 1  i 


GENERAL  OUTLINE.  [2 

data  necessary  for  these  observations  were  discussed  in  Prof. 
SAFFORD'S  paper,  Vol.  n,  pt.  n,  which  paper  was  prepared 
for  this  purpose.  In  the  case  of  the  Greenwich  observations 
from  1835  onward,  it  was  deemed  sufficient  to  apply  constant 
corrections  to  the  Eight  Ascensions,  determined  from  time  to 
time  by  comparisons  of  the  adopted  Eight  Ascensions  with 
the  standard  ones.  In  the  case  of  the  Declinations,  Boss's 
special  tables  were  used,  but  in  the  later  years  it  was  judged 
sufficient  to  apply  the  constant  correction  necessary  for  reduc- 
tion to  Boss's  standard. 

Palermo. — PIAZZT'S  observations  of  the  Sun  and  Planets  were 
completely  re-reduced,  the  zero  point  of  his  instrument  being 
determined  from  the  observed  Declinations. 

Paris. — LEVERRIER'S  reduction  of  the  Paris  observations 
from  1801  onward  was  made  use  of,  applying  the  correction 
necessary  to  reduce  the  results  to  the  adopted  standard. 

Konigsberg. — BESSEL'S  clock  corrections  were  individually 
corrected  by  the  new  positions  of  the  fundamental  stars,  so 
that  practically  the  Eight  Ascensions  may  be  considered  as 
completely  re-reduced. 

In  the  case  of  the  other  observatories,  it  was  deemed  suffi- 
cient to  determine,  by  a  comparison  of  the  adopted  or  of  the 
concluded  Eight  Ascensions  and  Declinations  of  the  funda- 
mental stars  with  the  standard  catalogue,  what  common  cor- 
rections were  necessary  for  reduction  to  the  standard.  When, 
however,  the  period  was  covered  by  Boss's  tables,  the  correc- 
tion which  he  gives  as  varying  with  the  Declination  was  ap- 
plied. After  more  mature  consideration,  I  am  inclined  to  think 
it  would  have  been  better  to  apply  a  constant  correction  to  the 
Declinations  in  every  case,  except  those  where  the  change 
with  the  Declination  was  quite  large. 

Although  these  processes  were  somewhat  heterogeneous,  it 
is  believed  that  the  main  object  of  referring  the  Declinations 
to  a  system  of  which  the  error  would  be  a  uniformly  varying 
quantity  was  fairly  well  attained.  The  subsequent  determi- 
nation of  this  error  both  in  Eight  Ascension  and  Declination 
is  a  necessary  part  of  the  work. 


3]  OBSERVATIONS  USED.  3 

The  following  is  a  list  of  the  observatories  whose  observa- 
tions of  the  Sun  and  Planets  were  included  in  the  work: 

Greenwich 1750-1892 

Palermo 1791-1813 

Paris __--  1801-1889 

Konigsberg 1814-1845 

Dorpat 1823-1838 

Cambridge . 1828-1844 

Berlin 1838-1842 

Oxford,  Radcliffe 1840-1887 

Pulkowa 1842-1875 

Washington  _ 1846-1891 

Leiden 1863-1871 

Strassburg 1884-1887 

Cape  of  Good  Hope 1884-1890 

The  number  of  the  meridian  observations  of  the  Sun,  and 
of  the  planets  Mercury,  Venus,  and  Mars,  actually  included  in 
the  work  is  approximately  as  follows: 

The  Sun 40,176 

Mercury  _' 5»421 

Venus 12,  319 

Mars 4,  114 

Total 62,030 

Semidiameters  of  Mercury  and  Venus. 

3.  The  reduction  of  the  semidiarneter  of  the  planets  was  a 
point  to  which  special  attention  was  given.  In  the  case  of 
Mercury,  the  adopted  semidiameter  at  distance  unity  was  3".34. 
The  values  adopted  by  the  various  observatories  in  reducing 
their  observations  varied  so  little  from  this  that  in  cases  where 
the  original  reductions  were  accepted  no  correction  was  applied 
for  the  difference.  So,  also,  when  the  observers  applied  a  cor- 
rection for  reducing  the  observed  center  of  light  to  the  actual 
center  of  the  planet,  no  revision  of  this  reduction  was  made. 
Such  was  supposed  to  be  the  case  with  the  Paris  observations. 
When  the  published  Eight  Ascension  was  that  of  the  center 
of  light  simply,  a  reduction  to  the  true  center  was  computed 
by  the  empirical  formula  used  in  the  Washington  observations. 
If  we  put  i  for  the  angle  between  the  Earth  and  Sun  as  seen 
from  the  planet,  then  1  -f-  cos  i  will  represent  the  fraction  of 


4  GENERAL   OUTLINE.  [3 

the  apparent  transverse  diameter  of  the  planet  that  is  illu- 
minated by  the  Sun.  It  was  assumed  that  when  the  illumina- 
tion was  such  that  the  thickness  of  the  crescent  approached 
zero,  the  point  observed  would  be  two-thirds  of  the  way  from 
the  center  of  the  planet  to  the  limb,  and  that  when  the  planet 
was  dichotomized  the  center  of  observation  would  be  five- 
twelfths  of  the  way  from  the  center  to  the  limb.  These  con- 
ditions, with  the  added  one  that  when  the  planet  was  fully 
illuminated  the  correction  should  vanish,  suggested  the  em- 
ployment of  the  formula 

Correction  =  seinidiameter  x  (1-cos  ^(5-f  cos  i) 

This  correction  was  to  be  multiplied  by  the  sine  or  cosine  of 
the  angle  which  the  line  of  cusps  made  with  the  meridian  to 
reduce  it  to  Right  Ascension  and  Declination  respectively. 

The  correction  being  practically  the  same  whenever  the 
Earth  and  planet  return  to  the  same  positions  in  anomaly,  it 
is  possible  to  embody  it  in  a  table  of  two  arguments,  one 
depending  on  the  longitude  of  the  Earth,  the  other  on  that  of 
the  planet.  Actually,  however,  the  table  was  arranged  in  a 
more  convenient  form,  in  which  one  argument  is  the  date  at 
which  Mercury  last  passed  perihelion,  and  the  other,  its  mean 
anomaly.  Owing  to  the  importance  which  this  correction  may 
assume,  a  partial  transcript  of  the  table  actually  employed  for 
the  reduction  in  Right  Ascension  is  given  on  the  next  page. 
Read  horizontally,  the  numbers  show  the  corrections  of  the 
argument  through  one  revolution  of  the  planet.  Vertically, 
they  may  be  regarded  as  giving  the  successive  corrections  corre- 
sponding to  any  one  position  of  the  planet,  while  the  Earth 
goes  through  a  complete  revolution.  The  table  as  actually 
used  extended  to  every  10°,  but  the  values  for  every  00°  of 
mean  anomaly  will  suffice  to  show  the  general  magnitude  of 
the  correction. 

The  correction  to  the  Declination  was  embodied  in  a  similar 
table,  which  it  is  not  deemed  necessary  to  print  at  present. 

In  the  case  of  Venus,  it  seems  scarcely  possible  to  decide 
upoiT  a  value  of  the  semidiauieter,  or  a  law  of  its  apparent 
change,  which  should  apply  to  all  parts  of  the  orbit.  After  a 


3] 


SEMIDIAMETERS   OF   MERCURY   AND   VENUS. 


careful  examination  of  the  data,  it  was  decided  to  reduce  all 
the  observations  with  the  semidiameter 

8-^-5+0".20 

when  made  with  modern  instruments,  and  to  use  a  value  0".3 
greater  in  earlier  observations.     The  actual  reductions  of  all 

Correction  for  defective  illumination  of  Mercury  in  R.  A. 
Arguments:  Date  of  perihelion  passage  at  side,  and  mean 
anomaly  "g"  at  top. 


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6  GENERAL  OUTLINE.  [4 

the  principal  series  of  observations  were  corrected  to  this  value 
of  the  element  in  question. 

Observations  of  the  estimated  center  of  Venus,  when  made 
more  than  one  hundred  days  from  superior  conjunction,  were 
rejected  altogether;  when  made  within  that  limit,  the  point 
observed  was  assumed  to  be  the  center  of  gravity  of  the  illu- 
minated portion  of  the  disk,  considered  as  a  plane  figure,  and 
the  necessary  reduction  to  the  center  was  always  applied. 

A  similar  correction  was  applied  to  observations  of  the  esti- 
mated center  of  Mars.  The  Paris  results,  after  1830,  and  the 
later  Greenwich  and  Washington  results,  are  published  with 
the  reduction  for  center  of  light  already  applied,  and  in  these 
cases  the  published  corrections  were  not  changed. 

Tabular  places. 

4.  The  tabular  elements  of  the  planets  adopted  for  correc- 
tion were  those  of  LEVERRIER'S  tables.  These  tables  having 
been  continuously  used  in  Astronomical  Ephemerides  since 
1864,  it  was  judged  more  convenient  to  adopt  the  theory  on 
which  they  were  based  as  the  provisional  one  to  be  corrected 
than  it  was  to  construct  a  new  provisional  theory.  As  the  tables 
in  their  original  form  are  extremely  cumbrous  to  use,  the 
theory  was  partially  reconstructed  by  making  manuscript 
tables  of  the  principal  perturbations,  which  were,  however, 
carried  only  to  tenths  of  seconds.  With  these  tables  the 
places  of  the  planets  were  computed  for  dates  previous  to  1864. 

As  places  of  the  Sun  were  necessary  not  only  for  direct  com- 
parison with  observations  of  the  Sun,  but  also  for  the  geocen- 
tric places  of  the  planets,  an  ephemeris  of  the  Sun's  longitude 
and  radius  vector  was  prepared  for  the  entire  period  1750-1864 
to  every  fifth  day,  the  lunar  perturbation  being  omitted  and 
afterward  applied  for  each  date  when  required. 

The  method  of  deriving  the  final  tabular  places  varied  with 
circumstances.  When  there  was  no  accurate  ephemeris  avail- 
able for  comparison,  which  was  the  case  before  1830,  it  was 
necessary  to  compute  a  completely  independent  set  of  tabular 
geocentric  places.  Sometimes  these  places  were  computed  for 
the  moment  of  the  individual  observations,  but  more  generally, 
when  the  observations  occurred  in  groups,  an  ephemeris  was 


4]  TABULAR  PLACES.  7 

computed  in  order  that  the  work  might  be  checked  by  differ- 
ences. After  1830  it  was  common  to  compute  an  ephemeris 
for  intervals  of  three,  five,  or  ten  days,  thus  deriving  the  cor- 
rections necessary  to  reduce  the  published  ephemerides  of  the 
Berliner  Jahrbuch  or  of  the  Nautical  Almanac  to  those  derived 
from  LEVERRIER'S  tables. 

Until  this  plan  was  mapped  out,  and  work  well  in  progress 
upon  it,  it  was  not  noticed  that  the  planetary  masses  adopted  in 
LEVERRIER'S  tables  were  so  diverse  that  corrections  to  reduce 
the  geocentric  places  to  a  uniform  system  of  masses  would  be 
necessary.  Although  theoretically  the  necessary  reductions 
were  very  simple,  I  can  not  but  feel  that  the  application  of 
such  corrections  involves  more  or  less  doubt  and  uncertainty, 
and  that  it  would  have  been  better  to  have  constructed  pro- 
visional tables  based  on  uniform  masses  quite  independent  of 
those  of  LEVERRIER. 

In  Annales  de  V  Observatoire  de  Paris,  Vol.  n,  LEVERRIER 
gives  the  following  values  of  the  masses  used  by  him  as  the 
basis  of  his  provisional  theory : 

Mercury  .      .  — ^  =  .000  000  333  .  . 

Venus 40T847  =-0000024885 

Earth  .  QKA .ftQg    =.000002 


Mars 2  680  337  =  -000  00°  373 

The  following  table  shows  the  factors  by  which  these  masses 
were  multiplied  in  the  cases  of  the  several  planets  in  LEVER- 
RIER'S final  tables.  They  were  controlled  by  induction  from 
the  numbers  of  the  tables  themselves,  the  result  of  which  was 
found  in  all  cases  to  agree  with  the  statements  in  the  introduc- 
tion to  the  tables. 

In  the  last  line  of  the  table  is  shown  the  factor  used  in  the 
present  provisional  theory. 


GENERAL  OUTLINE. 


Mercury. 

Venus. 

Earth. 

Mars. 

In  tables  of  — 
The  Sun  . 

i  004. 

o  SCK 

Mercury  

j 

Venus  

i 

j 

j 

Mars 

o.  071; 

i.  0026 

Present  work 

j 

i 

o  86s? 

As  in  the  actual  work  the  masses  of  Mercury  and  Venus 
were  to  be  determined  from  the  observed  periodic  perturba- 
tions which  they  produced,  it  was  necessary  that  the  perturba- 
tions produced  by  them  should  all  be  carefully  reduced  to  the 
adopted  standard.  The  reduction  was  less  necessary  in  the 
case  of  Mars,  but  was  carried  through  all  the  work  relating  to 
the  Sun. 

Comparison  of  observations  and  tables. 

5.  The  result  of  each  separate  observation  of  each  body  was 
compared  with  the  tabular  result  thus  derived.    The  residuals 
were  then   taken   and   divided   into   groups.    The   interval 
between  the  extreme  dates  of  each  group  was  always  taken 
so  short  that  it  could  be  presumed  that  the  mean  of  all  the 
residuals  would  be  the  correction  for  the  mean  of  all  the  dates. 
The  general  rule  was  that  the  interval  should  not  exceed  four 
or  five  days  in  the  case  of  Mercury,  or  six  or  eight  days  in 
that  of  Venus,  and  that  not  more  than  six  or  eight  observa- 
tions should  be  included  in  a  single  group.     In  taking  these 
means,  weights  were  assigned  to  the  results  of  each  observa- 
tory founded  on  the  discordance  of  its  residuals.    Then  to  each 
mean  a  weight  was  again  assigned  equal  to  the  sum  of  the 
weights  of  the  individual  residuals  when  these  were  few  in 
number,  but  not  allowed  to  exceed  a  certain  limit,  how  great 
soever  might  be  the  sum  of  the  individual  weights. 

Equations  of  condition. 

6.  Each  mean  result  thus  derived  formed  the  absolute  term 
of  an  equation  of  condition  for  correcting  the  tabular  elements. 
The  number  of  these  equations  was  as  follows: 

Equations. 

The  Sun . 11,676 

Mercury ___ 3,  929 

Venus 4,849 

Mars i,  597 


6]  EQUATIONS   OF  CONDITION.  9 

In  forming  the  equations  of  condition  from  observations  of 
the  planets,  I  adopted  the  system  suggested  in  the  introduc- 
tion to  Vol.  i  of  these  publications,  namely,  the  determination 
of  the  solar  elements  not  only  from  observations  of  the  Sun 
itself,  but  from  observations  of  each  of  the  planets.  The  reason 
for  this  course  is  quite  simple  and  obvious.  An  observation  of 
the  position  of  a  planet  as  seen  from  the  Earth  is  the  exact 
equivalent  of  an  observation  of  the  Earth  as  seen  from  a 
planet,  and  thus  depends  equally  upon  the  elements  of  both 
orbits.  Hence,  whatever  elements  of  the  Earth's  orbit  could 
be  determined  by  observations  made  from  a  planet  can  equally 
be  determined  by  observations  made  upon  the  planet.  A 
strong  reason  for  proceeding  upon  this  plan  was  found  in  the 
very  large  errors,  both  accidental  and  systematic,  to  which 
observations  of  the  Sun  are  liable. 

The  advantages,  however,  have  not  proved  relatively  so 
great  as  were  anticipated.  The  eccentricity  and  perihelion  of 
the. Earth's  orbit  come  out  in  the  solution  of  the  normal  equa- 
tions as  functions  of  those  of  the  planetary  orbit  to  so  great  an 
extent  that  their  weight  is  much  less  than  that  which  would 
correspond  to  independent  determinations  from  the  same  num- 
ber of  observations.  On  the  other  hand,  the  determination 
of  these  elements  from  observations  of  the  Sun  proved  to  be 
much  more  consistent  than  was  expected,  thus  indicating  a 
high  degree  of  precision. 

The  case  is  different  with  the  Sun's  mean  longitude  referred 
to  the  Stars.  Here  systematic  and  personal  errors  enter  so 
largely  that  the  results  from  Mercury  and  Venus  appear  to  be 
rather  more  reliable  than  those  from  the  Sun  itself.  In  the 
case  of  these  planets  it  fortunately  happens  that  the  weight  of 
the  result  derived  for  the  Sun's  mean  longitude  is  not  mate- 
rially diminished  by  the  uncertainty  of  the  corresponding 
element  of  the  planet,  the  errors  of  the  two  mean  longitudes 
being  nearly  separated  in  a  series  of  observations  equally  dis- 
tributed around  the  orbit. 

The  systematic  errors  in  observations  of  the  Sun  rendered 
it  unadvisable  to  determine  the  elements  of  the  Earth's  orbit 
from  observations  of  the  Sun  by  a  single  system  of  equations. 
The  solar  observations,  therefore,  were  classified  according  to 


10  GENERAL   OUTLINE.  [6 

the  observatory  where  made,  and  divided  into  periods  rarely 
exceeding  eight  years  in  length.  The  elements  are  separately 
derived  from  the  observations  of  each  period.  This  system  has 
the  advantage  of  eliminating  to  a  large  extent  the  injurious 
effect  of  systematic  and  personal  error  upon  the  eccentricity 
and  perihelion  of  the  Earth's  orbit,  and  also  enabling  us  to 
judge  of  the  precision  of  the  corrections  to  those  elements  by 
the  discordance  among  separate  results. 

Meridian  observations  of  the  Sun  and  Planets  are  referred 
to  the  fundamental  stars,  while  the  Eight  Ascensions  of  the 
latter  are  referred  to  the  equinox,  the  position  of  which  has 
heretofore  depended  on  observations  of  the  Sun.  The  adopted 
position  of  the  fundamental  stars  therefore  comes  in,  to  a  cer- 
tain extent,  as  the  basis  of  the  work,  and  the  constant  parts 
of  their  systematic  corrections  are  among  the  results  to  be 
derived. 

Thus,  in  the  case  of  the  equations  pertaining  to  the  three 
planets,  the  following  corrections  were  introduced  as  unknown 
quantities  : 

Correction  of  the  mass  of  Mercury  or  of  Venus. 

Corrections  to  the  elements  of  the  orbit  of  the  planet 
observed. 

Correction  of  the  obliquity  of  the  ecliptic. 

Corrections  to  the  Sun's  mean  longitude,  eccentricity,  and 
longitude  of  perihelion. 

Common  corrections  to  the  adopted  Eight  Ascensions  and 
Decimations  of  the  fundamental  stars. 

In  the  case  of  Mercury  an  adopted  hypothetical  correction 
of  the  ratio  of  the  radius  vector  of  the  planet  to  that  of  the 
Earth  was  also  included  in  the  equations,  although  little  doubt 
could  be  felt  that  the  true  value  of  such  a  quantity  must  be 
zero.  The  reason  for  introducing  it  will  be  explained  here- 
after. 

Determinations  of  the  masses  and  secular  variations. 

7.  The  secular  variation  of  all  the  preceding  elements,  the 
mean  distances  excepted,  was  also  introduced  into  the  equa- 
tions from  observations  of  the  planets.  In  addition  to  the 
above  elements,  the  mass  of  Venus  appeared  in  the  equations 


7]  MASSES  AND   SECULAR  VARIATIONS.  11 

derived  from  observations  of  the  Sun,  Mercury,  and  Mars,  and 
the  mass  of  Mercury  in  the  equations  derived  from  obser- 
vations of  Venus.  The  coefficients  of  the  masses,  however, 
depended  wholly  upon  the  periodic  perturbations. 

Were  it  quite  certain  that  the  secular  variations  arise 
wholly  from  the  masses  of  the  known  planets,  the  masses 
could  of  course  be  derived  from  these  variations,  and  the  lat- 
ter would  appear  in  the  equations  of  condition  only  through 
the  mass  itself.  On  this  hypothesis  the  secular  variations 
would  not  appear  in  the  equations,  but  only  the  masses.  But 
it  is  well  known  that  the  perihelion  of  Mercury  is  subject  to  a 
secular  variation  which  can  not  be  accounted  for  by  any  ad- 
missible masses  of  the  known  disturbing  planets.  The  same 
thing  may  well  be  true  of  the  secular  variations  of  the  other 
elements.  It  is  therefore  necessary,  in  the  absence  of  a  known 
cause  for  such  deviations,  to  derive  the  masses  of  the  planets 
independently  of  the  secular  variations.  In  the  case  of  Mars 
the  mass  is  obtained  with  all  necessary  precision  from  the  sat- 
ellites. It  is,  however,  different  in  the  case  of  Mercury  and 
Venus.  Here  no  resource  is  left  us  but  to  determine  them 
from  the  periodic  inequalities.  As  the  inequality  produced  by 
Venus  in  the  Earth's  longitude  is  rarely  more  than  eight  sec- 
onds, it  might  seem  that  the  coefficient  would  be  too  small  to 
obtain  a  sufficiently  precise  value  of  the  mass.  But  in  the 
case  of  observations  upon  the  Sun,  Mercury,  and  Mars  the 
error  of  the  determination  of  the  mass  in  question  may  be 
almost  indefinitely  reduced  by  multiplication  and  extension 
of  the  observations  without  danger  of  systematic  error. 

To  illustrate  this,  let  us  suppose  the  Sun's  longitude  to  be 
determined  with  a  meridian  instrument  only  once  a  year,  say 
at  equal  intervals  of  three  hundred  and  s^xty-five  days.  Let 
the  longitudes  thus  observed  be  compared  with  an  ephemeris 
in  which  the  elements  are  affected  with  only  slight  errors. 
Leaving  out  of  consideration  the  periodic  perturbations  pro- 
duced by  the  planets,  the  comparison  of  the  observed  longi- 
tudes with  the  tabular  ones  through  an  entire  century  should 
be  nearly  constant.  Any  error  affecting  all  the  longitudes 
alike  would  appear  as  a  constant.  The  errors  of  mean  motion 


12  GENERAL   OUTLINE.  [7 

would  vary  imiformly  with  the  time.  Thus  the  other  elements 
would  be  nearly  constant,  and  could  be  still  more  approxi- 
mately represented  by  a  slight  apparent  secular  variation. 

Now  let  the  disturbing  action  of  a  planet,  say  Venus,  be  in- 
troduced. We  should  then  have  a  series  of  deviations  from  the 
law  of  uniform  increase,  which  would  enable  us  to  evaluate 
the  mass  of  the  planet.  The  value  of  this  mass  thus  derived 
would  not  be  affected  by  any  systematic  error  common  to  all 
the  observations,  nor  even  by  such  an  error  which  varied  uni- 
formly with  the  time.  Nor  would  small  errors  in  the  adopted 
elements  of  the  Sun  have  any  effect  upon  the  result. 

If  this  would  be  the  case  for  observations  made-  only  at  a 
certain  point  of  the  orbit,  a  fortiori  would  it  be  the  case  for 
the  observations  made  at  various  points  of  the  orbit,  since  any 
tendency  to  a  systematic  effect  of  the  errors  of  observation 
would  thereby  be  ultimately  eliminated. 

Considerations  almost  identical  apply  to  the  case  of  observa- 
tions upon  either  of  the  planets  when  we  consider  the  action 
of  the  other  planet  upon  the  planet  observed  and  upon  the 
earth.  But  they  do  not  apply  to  the  case  of  the  action  of  the 
earth  itself  upon  the  observed  planet,  or  vice  versa.  For  ex- 
ample, in  the  case  of  observations  of  Venus,  we  may  suppose 
that  all  observations  made  when  Venus  is  at  a  certain  point 
of  its  relative  orbit,  near  inferior  conjunction,  say  one  month 
before  inferior  conjunction,  are  affected  with  a  certain  error 
common  to  all  observations  made  at  that  point  of  the  orbit. 
Since  the  perturbations  produced  by  the  third  planet  will  in 
the  long  run  have  all  values,  positive  and  negative,  for  these 
several  observations,  the  systematic  error  in  question  will  not 
affect  the  ultimate  value  of  its  mass.  But  the  perturbations 
of  Venus  produced  by  the  Earth,  as  well  as  those  of  the  Earth 
produced  by  Venus,  will  not  have  all  values  in  such  a  case,  but 
only  special  ones  dependent  on  the  relative  position.  Hence, 
determinations  of  these  masses  might  be  affected  by  errors  of 
the  kind  in  question.  We  conclude,  therefore,  that  the  mass 
of  the  Earth  can  not  be  satisfactorily  determined  by  the  peri 
odic  perturbations  which  it  produces  in  the  motion  of  any 
planet,  nor  that  of  Venus  by  observations  on  Venus  through 
its  periodic  perturbations  of  the  Earth. 


8]  TRANSITS  OF  VENUS  AND  MERCURY.  13 

In  the  solution  of  the  equations  of  condition  the  method  of 
least  squares  has  been  used  throughout,  the  arrangement  of 
the  work,  the  choice  of  quantities  to  be  corrected,  and  the 
accuracy  of  the  coefficients  being  so  chosen  as  to  minimize  the 
great  mechanical  labor  of  making  the  necessary  multiplica- 
tions. The  adoption  of  this  method  was  necessary  in  order  to 
separate,  so  far  as  possible,  the  various  unknown  quantities 
and  show  to  what  extent  their  values  were  interdependent. 
By  no  other  method  of  combination  could  so  large  a  number 
of  unknown  quantities  have  been  separately  determined  in  a 
way  which  would  have  been  at  all  satisfactory.  On  the  other 
hand,  in  combining  the  final  results  and  deciding  upon  the 
values  of  the  corrections  to  be  adopted,  the  method  has  not 
always  been  applied,  for  reasons  which  will  be  developed  in 
Chapter  IY. 

Introduction  of  results  of  observations  on  transits  of  Venus  and 

Mercury. 

8.  In  the  case  of  Mercury  and  Venus  the  observed  transits 
over  the  Sun  give  relations  between  the  corrections  to  the 
elements  more  accurate  than  those  ordinarily  derivable  from 
meridian  observations.  This  is  especially  the  case  with  Venus. 
The  value  of  these  observations  is  greatly  increased  by  the 
fact  that  they  are  made  when  the  planet  is  near  inferior  con- 
junction, and  therefore  nearest  to  the  Earth,  and  in  a  point  of 
the  relative  orbit  where  meridian  observations  are  necessarily 
most  uncertain.  In  the  case  of  Venus  the  error  of  the  helio- 
centric place  will  be  more  than  doubled  in  the  case  of  the  geo- 
centric place  during  a  transit.  As,  however,  the  observation 
of  a  transit  gives  no  one  element,  but  only  an  equation  of  con- 
dition between  the  values  of  all  the  elements  at  the  epoch,  the 
only  way  of  treating  it  is  to  introduce  the  result  as  such  an 
equation,  with  its  appropriate  weight.  The  determination  of 
the  proper  weight  is  a  difficult  matter.  The  systematic  errors 
of  meridian  observations  are  such  that  the  theoretical  value 
of  the  weights  assignable  to  so  great  a  mass  as  we  have  dis- 
cussed would  be  entirely  illusory.  In  fact  so  great  is  the 
weight  assignable  to  the  observed  transits  of  Venus  that  if 
we  should  regard  the  results  of  each  transit  as  a  condition  to 


14  GENERAL   OUTLINE.  [8 

be  absolutely  satisfied  we  should  not  be  dangerously  in  error. 
I  conclude,  therefore,  that  there  is  more  danger  of  assigning 
too  small  than  too  great  a  weight  to  these  observations. 

In  order  to  determine  what  change  was  produced  in  the  re- 
sults by  the  use  of  the  observed  transits  over  the  sun's  disk, 
two  separate  solutions  of  the  equations  of  condition  for  Mer 
cury  and  Venus  were  made.  In  the  one,  termed  solution  A, 
the  meridian  observations  alone  were  used;  in  the  other, 
termed  solution  B,  the, combined  equations  formed  by  adding 
the  normal  equations  derived  from  the  transits  to  those  given 
by  the  meridian  observations  were  used. 

In  the  case  of  solution  A  it  was  originally  supposed  that  by 
using  the  mean  epoch  of  all  the  observing  in  the  case  of  each 
planet  as  that  from  which  the  time  was  to  be  reckoned,  the 
normal  equations  for  the  secular  variations  would  be  almost 
completely  separated  from  those  for  the  corrections  to  the 
elements  themselves.  The  separation  would  be  complete  were 
the  observations  at  different  epochs  similarly  distributed 
around  the  orbit.  But,  as  a  matter  of  fact,  it  was  found  that 
the  accidental  deviations  from  this  symmetry  were  so  consider 
able  that  the  separation  could  not  be  regarded  as  complete. 
The  solution  was  therefore  made  by  successive  approximations, 
the  terms  depending  on  the  secular  variations  being  in  the 
first  approximation  dropped  from  the  normal  equations  for  the 
corrections  to  the  elements,  and  afterwards  included  when 
approximately  determined,  and  vice  versa. 

In  the  case  of  solution  B,  in  which  the  transits  were  included, 
such  a  separation  did  not  occur,  and  the  equations  were  solved 
in  the  usual  rigorous  way  for  all  the  unknown  quantities. 


CHAPTER  II. 

DISCUSSION    AND    RESULTS    OF    OBSERVATIONS   OF    THE 

SUN. 

Treatment  of  the  Eight  Ascensions. 

9.  The  meridian  observations  of  the  Sun  have  been  treated 
on  a  system  different  in  some  points  from  that  adopted  in  the 
case  of  the  planets.  It  was  possible  to  simplify  the  treatment 
by  supposing  that  the  small  latitude  of  the  Sun  was  always  a 
definitely  known  quantity,  so  that  when  the  observations  were 
corrected  for  it  the  apparent  motion  of  the  Sun  could  be  sup- 
posed to  take  place  along  the  great  circle  of  the  ecliptic.  This 
allowed  the  correction  of  the  elements  to  depend  on  but  two 
quantities — the  obliquity  of  the  ecliptic  and  the  Sun's  true 
longitude.  Assuming  the  obliquity  to  be  known,  the  longi- 
tude of  the  Sun  could  always  be  determined  from  an  observa- 
tion of  its  Right  Ascension.  An  observed  Eight  Ascension 
being  compared  with  a  tabular  one,  the  residual  gives  rise  to 
an  equation  of  condition  between  the  correction  of  the  long- 
itude, A,  of  the  obliquity,  £,  and  of  the  Right  Ascension  of  the 
Sun,  a: 

da  =  cos  s  sec2  ddX  —  £  tan  e  sin  2ad?. 

This  equation  may  be  used  to  express  the  error  of  the  longi- 
tude in  terms  of  the  error  of  the  obliquity  and  of  the  Right 
Ascension  as  follows : 

#A  =  sec  8  cos2  dda  +  J  tan  e  sin  2Xds 
=  sec  s  cos2  6$a  +  0.21  sin  2Xds 

The  elements  mainly  to  be  determined  from  the  observations 
in  Right  Ascension  being  the  eccentricity  and  perihelion  of 
the  Earth's  orbit,  each  of  the  coefficients  of  which  go  through 
a  period  in  a  year,  the  effect  of  the  small  term  —  0.21  $s  sin  2A 
whose  coefficient  does  not  amount  to  0".10  after  1800,  and  has 
a  period  of  half  a  year,  will  be  practically  without  influence 

15 


16  OBSERVATIONS  OF   THE   SUN.  [10 

on  the  result.  The  system  was  therefore  adopted  of  deriving 
the  residual  in  longitude  directly  from  the  residual  in  Eight 
Ascension  by  the  formula 


where 

F  =  cos2  3  sec  e  . 

The  residual  #A  in  true  longitude  is  then  to  be  expressed  in 
terms  of  the  residual  61"  in  mean  longitude  and  of  corrections 
to  the  eccentricity  and  to  the  longitude  of  the  perigee  relative 
to  the  Stars.  In  this  expression  the  coefficient  of  the  residual 
in  mean  longitude  was  always  taken  as  unity,  the  value  of  the 
correction  being  so  small  in  the  case  of  LEVERRIER'S  tables 
that  no  appreciable  error  would  result  from  this  supposition. 
Thus  each  residual  in  Eight  Ascension  would  give  rise  to  an 
equation  of  condition  of  the  form  — 


61"  +  Pe"6n"  +  E6e"  =  tfA  =  ~F6a 

We  are  here  to  regard  61"  and  6n"  as  corrections  to  the 
Eight  Ascensions  relative  to  the  clock  stars,  and  not  to  the 
Sun's  longitude  or  perigee  simply.  I  shall  therefore  use  the 
symbol  c  instead  of  61"  to  express  the  relative  correction  here- 
after. 

Treatment  of  the  Declinations. 

10.  The  declination  of  the  Sun  in  the  case  supposed  is  a 
function  only  of  the  longitude  and  obliquity.  The  equation 
for  expressing  the  observed  correction  in  Declination  in  terms 
of  the  corrections  to  these  two  quantities  is 

Ad  =  sin  a6s  -f  cos  a  sin  sSX 

Thus  each  observation  of  the  Sun's  Declination  gives  rise  to 
an  equation  of  condition  of  this  form. 

It  is  however  to  be  supposed  that  the  observations  in  Decli- 
nation made  at  each  observatory  will  be  affected  by  a  constant 
error.  If  the  observations  are  truly  reduced  to  the  standard 
system  of  star  places,  this  error  will  be  that  of  the  standard 
system.  As  a  matter  of  fact,  however,  observations  made  in 
the  daytime,  especially  on  the  Sun  and  at  noon,  are  made 
under  circumstances  so  different  from  night  observations  on 


11]       FORMATION  OF  EQUATIONS  IN  RIGHT  ASCENSIONS.         17 

stars  that  we  can  not  assume  the  error  of  the  reduced  declina- 
tion to  be  necessarily  the  same  as  that  of  the  star  system. 
We  must,  therefore,  in  each  case,  regard  the  constant  error  in 
declination  as  something  peculiar  to  the  observatory  and  the 
instrument,  which  may  or  may  not  be  worthy  of  subsequent 
discussion.  Thus  each  residual  in  declination  gives  rise  to 
an  equation  of  condition, 

j#o  4.  cos  a  sin  £(U  -f  sin  ade  =  AS 

Ad  being  the  excess  of  observed  over  tabular  declination, 
and  AdQ  the  common  error  of  all  the  measured  declinations  of 
any  one  series. 

Formation  of  the  equations  from  Right  Ascensions* 

11.  The  method  of  treating  the  observed  Eight  Ascensions 
of  the  Sun  was  suggested  by  the  fact  that  they  are  peculiarly 
liable  to  systematic  and  personal  errors ;  the  former  likely  to 
change  with  the  seasons,  and  to  be  different  for  different  in- 
struments; and  the  latter  to  continue  through  the  work  of  one 
observer.  It  is  now  well  understood  that  the  observed  Eight 
Ascensions  of  the  mean  of  the  Sun's  two  limbs  relative  to  the 
fixed  stars  are  affected  by  personal  errors,  no  means  of  elimi- 
nating which  have  yet  been  tried.  In  a  series  of  observations 
made  by  a.  single  observer,  under  uniform  conditions,  this  error 
would  systematically  affect  only  the  relative  mean  of  the  Eight 
Ascensions  of  the  Sun  and  Stars,  leaving  the  eccentricity  and 
perigee  derived  from  the  observations  substantially  correct. 

On  taking  up  the  work  it  was  also  supposed  that,  owing  to 
the  different  effect  of  the  Sun's  rays  upon  the  instrument  at 
different  seasons,  and  the  different  circumstances  under  which 
observations  were  made,  the  Eight  Ascensions  of  the  Sun 
would  be  affected  by  errors  varying  in  a  regular  way  through 
the  year,  but  not  wholly  expressible  as  a  term  of  single  annual 
period.  It  was  therefore  deemed  best  to  consider  the  observa- 
tions possibly  affected  by  an  error  of  double  period,  having  the 
form 

x'  cos  2g  -f  y'  sin  2g 

5690  N   ALM 2 


18  OBSERVATIONS  OF  THE   SUN.  [11 

The  introduction  of  the  coefficients  x1  and  y'  added  two  more 
terms  to  the  equations  of  condition,  which  terms,  however,  did 
not  express  any  astronomical  fact,  but  only  the  possible  errors 
of  the  observations. 

An  additional  and  very  important  element  to  be  determined 
from  the  observed  Eight  Ascensions  was  the  mass  of  Venus. 
The  question  now  arose  whether,  by  a  uniform  series  of  obser- 
vations, extending  through  some  definite  period,  the  correc- 
tions to  the  eccentricity  and  perigee  and  the  coefficients  x1  and 
y1  could  be  completely  separated  from  the  coefficients  of  the 
correction  to  the  mass  of  Venus.  Examination  showed  that 
from  such  a  series  of  observations,  extending  through  eight 
years,  the  mass  of  Venus  could  be  determined  irrespective  of 
all  systematic  errors  repeating  themselves  with  the  season, 
provided  that  the  observations  were  equally  distributed 
throughout  the  year,  or  even  that  an  equal  number  were  made 
at  the  same  time  through  successive  years.  As  neither  of 
these  conditions  are  practically  fulfilled  it  was  judged  best  to 
assume  in  the  beginning  that  the  systematic  errors  of  an  un- 
known kind  repeated  themselves  at  each  season  during  an 
eight-year  period,  and  that  they  could  be  expressed  in  the 
form 

c  +  x  cos  g  H-  y  sin  g  +  x1  cos.  2g  +  y'  sin  2g 

x  and  y  would  appear  as  errors  of  eccentricity  and  perigee 
which  could  not  be  eliminated. 

The  quantities  actually  introduced  as  the  uuknown  ones  of 
the  equations  of  condition  were  as  follows: 

X,  the  factor  of  correction  of  the  mass  of  Venus  j 
#,  one-fifth  the  correction  to  the  eccentricity; 
y,  one-fifth  the  correction  e"dn" ; 

x',y',  one-tenth  the  coefficients  expressing  the  .supposed 
error  of  double  period  arising  from  all  causes  whatever ; 

c,  the  constant  correction  to  the  Eight  Ascension  of  the 
Sun  relative  to  the  Stars. 

The  coefficient  of  c  was  supposed  unity  throughout.  The 
reduction  of  the  residual  in  Eight  Ascension  to  that  in  Longi- 
tude and  the  other  factors  were  taken  from  a  table  like  the 
following,  of  which  the  argument  was  the  day  of  the  year. 


• 
• 


11]        FORMATION  OF  EQUATIONS  IN  EIGHT  ASCENSION.          19 

Separate  tables  were  constructed  for  1802  and  1850,  but  they 
were  so  nearly  identical  that  no  distinction  need  be  made 
between  them.  Furthermore,  the  error  introduced  by  sup- 
posing the  mean  anomaly  to  have  the  same  value  on  the  same 
day  of  every  year  is  entirely  unimportant. 

Table  of  coefficients  for  expressing  errors  of  the  Sun's  Right 
Ascension  in  terms  of  errors  of  the  elements  of  the  Earth's 
orbit. 


da 
dl 

dl 

Coefficients  of  — 

da 

,=0^ 

*-*«* 

f 

y' 

Jan.         I  

1.09 

o.  91 

4-    O.I 

—  10.  0 

-[-  o.  i 

4-10.0 

II  

1.07 

0-93 

1.8 

9.8 

3-5 

9-4 

21  

1.04 

o.  96 

3-4 

9.4 

6-5 

7.6 

31  

I.  OI 

0.98 

8.7 

8.7 

Feb.      10  

0.98 

I.  OI 

6.'  4 

7.7 

9.8 

4-  \\ 

20  

o.  96 

.04 

+  7.6 

-6.5 

+  9-9 

-  1.6 

Mar.       2  

0.94 

.06 

8.6 

5.1 

8.7 

4.9 

12  

o.  92 

.08 

9.4 

3-5 

6.6 

7.5 

22  

o.  92 

.08 

9-8 

1.9 

3-7 

9-3 

Apr.        I  

°-93 

.07 

IO.  O 

—    O.  I 

+  0.3 

IO.  O 

II  

0.94 

.05 

+  9-9 

4-  1.6 

*•>         T 

o*  *• 

-  9-5 

21  

o.  96 

.  03 

9-5 

3-2 

6.  i 

7-9 

May        i  

0.99 

.  OI 

8.8 

4.8 

8.4 

5-4 

ii  

I.  02 

0.98 

7-8 

6.2 

9-7 

—    2.  2 

21  

1.05 

0.95 

6.6 

7-5 

9.9 

I.  2 

31  

1.07 

0.93 

+  5.3 

+  8.5 

-  8.9 

—  4-5 

June      10  

1.09 

o.  91 

3-7 

9-3 

6.9 

7.2 

20  

I.  10 

o.  91 

2.  I 

9.8 

4.1 

9.1 

30—- 

1.09 

0.91 

+  0.4 

IO.  O 

-  0.7 

10.  0 

July      10  

1.  08 

0.93 

9.9 

+  2.7 

9.6 

20  

1.05 

0.95 

—  3.0 

+  9-5 

+  5.8 

-    8.2 

30  

1.03 

0.07 

4.6 

8.9 

8,2 

5-7 

Aug.       9  

I.  OO 

.  OO 

6.  i 

8.0 

9-6 

+  2.7 

19  

0.97 

•  03 

7-3 

6.8 

10.  0 

—  0.8 

29  

°-95 

•  05 

8.4 

5-4 

9.1 

4.  i 

Sept.      8  

0-93 

.07 

—  9.2 

+  3«9 

+  7.2 

-6.9 

18 

0.92 

.08 

9-7 

2-3 

4.5 

8.9 

28.... 

o.  92 

.08 

10.  0 

+  0.6 

4-    1.2 

9-9 

Oct.        8... 

0-93 

.07 

9.9 

—  1.  1 

—    2.  2 

9-7 

18  

o-9S 

•  05 

9.6 

2.8 

5-4 

8.4 

28.... 

0.97 

I.  02 

—  9.0 

—  4.4 

—  7.9 

—  6.1 

Nov.      7  

I.  00 

0.99 

8.1 

5.9 

9-5 

-  3-1 

17  

1.03 

o.  96 

7.o 

7.2 

IO..O 

+  0.3 

Dec.       7~~" 

.06 
.08 

0.94 
o.  92 

5.6 
4-  i 

8.3 

9.1 

9-3 

7-5 

3.7 
6.6 

17  

.09 

o.  91 

-   2.  5 

—  9-7 

—  4-9 

+  8.7 

27.... 

.09 

o.  91 

-0.8 

—  IO.  O 

—  1.6 

+  9-9 

20  OBSERVATIONS   OF  THE   SUN.  [12 

Finally,  throughout  the  work  the  equations  of  condition 
were  expressed  only  in  entire  numbers,  the  decimals  being 
neglected.  To  lessen  the  number  of  equations  of  condition, 
the  residuals  were  divided  into  groups  generally  covering  from 
ten  to  fifteen  days,  the  length  of  the  group  being  determined 
by  the  condition  that  the  perturbations  of  Venus  must  not 
change  much  during  the  period. 

While  the  formation  and  solution  of  the  equations  of  condi- 
tion on  this  system  were  going  on,  it  was  found  that  the  intro- 
duction of  the  assumed  coefficients  x1  and  y'  was  a  refinement 
productive  of  little  or  no  good  result.  In  fact,  the  observa- 
tions of  the  Sun  proved  to  be  much  freer  from  annual  sources 
of  error  than  I  had  supposed,  as  will  be  seen  by  the  tables  of 
their  results  soon  to  be  given.  This  is  shown  by  the  general 
consistency  of  the  corrections  to  the  eccentricity  and  perigee 
given  by  the  work  at  the  same  or  different  observatories  dur- 
ing different  periods. 

In  marked  contrast  to  this  is  the  discordance  among  values 
of  the  correction  c  to  the  relative  Eight  Ascensions  of  the  Sun 
and  Stars.  This  quantity  it  is  that  is  affected  by  personal 
error  and  possibly  by  the  effect  of  the  Sun  on  the  instrument. 
Under  a  perfect  system  of  discussion  it  would  be  advisable  to 
determine  it  separately  for  each  observer.  This  however  was 
practically  impossible. 

Solution  of  the  equations. 

12.  For  the  purposes  of  forming  and  solving  the  normal 
equations,  the  equations  of  condition  were  divided  into  groups 
of  generally  from  four  to  eight  years,  the  exact  lengths  of 
which  will  be  seen  from  the  following  exhibit  of  results.  The 
equations  for  each  period  were  solved  on  the  supposition  that 
the  corrections  were  constant  during  the  period.  Thus  every 
separate  result  is  independent  of  every  other,  except  so  far  as 
they  may  depend  on  the  same  instrument  or  the  same  observer 
at  different  times. 

The  first  column  shows  the  years  through  which  the  obser- 
vations extend. 

The  second  one  shows  to  the  nearest  year  the  value  of  T — 
that  is,  the  fraction  of  the  century  after  1850. 


12]  SOLUTION  OF  THE  EQUATIONS.  21 

The  third  column  shows  the  value  of  //,  or  that  factor  which, 
being  multiplied  by  the  adopted  mass  of  Venus,  is  to  be  applied 
as  a  correction  to  that  mass,  to  obtain  the  value  given  by  the 
observations. 

All  systematic  errors  arising  from  the  instrument  and  the 
observer  are  so  completely  eliminated  from  the  separate  de- 
terminations of  X  that  they  may  be  regarded  as  absolutely 
independent  of  each  other,  that  is — as  not  affected  by  any 
common  systematic  error. 

We  have  next  the  relative  weight  assigned  to  each  value 
of  //,  which  is  determined  in  the  usual  way  from  the  solu- 
tion, and  is,  therefore,  on  a  different  scale  for  different  ob- 
servatories. 

Next  is  given  the  value  of  c,  or  the  apparent  correction  to 
the  Eight  Ascension  of  the  Sun,  relative  to  the  assumed  Eight 
Ascensions  of  the  Stars,  as  given  by  observations  during  the 
several  periods  and  expressed  in  seconds  of  arc,  followed  by 
the  weights  assigned  to  the  separate  results. 

The  next  two  columns,  the  corrections  to  the  solar  eccen- 
tricity and  to  the  longitude  of  the  perigee,  require  no  further 
explanation. 

Eespecting  the  weights  ultimately  assigned  to  these  quanti- 
ties, and  to  GJ  it  is  to  be  remarked  that  they  are  the  result  of 
judgment  more  than  of  computation.     It  is  only  possible  to 
enumerate  in  a  general  way  with  some  examples  the  consider 
ations  on  which  they  are  based. 

In  assigning  the  weight  of  c  the  number  of  observers  en- 
gaged is  an  important  factor  in  determining  it.  Other  factors 
are  the  steadiness  of  the  atmosphere  and  the  adaptation  of  the 
instrument  to  this  particular  work.  General  consistency  is 
an  important  factor  in  the  assignment.  In  this  respect  the 
Cambridge  observations  are  quite  remarkable  ;  if  their  excel- 
lence corresponds  to  their  consistency  they  must  be  the  best 
ones  made. 

It  will  be  seen  that  PIAZZI'S  results  are  thrown  out  en- 
tirely. The  wide  range  of  his  values  of  c  led  to  the  inquiry 
whether  more  consistent  results  would  be  obtained  by  taking 
shorter  periods,  but  it  was  found  that  the  values  of  c  varied 
from  time  to  time  in  such  an  irregular  way  that  his  instrument 


22 


OBSERVATIONS   OF   THE    SUN. 


[12 


must  have  been  affected  by  some  extraordinary  cause  of  error, 
unless  some  mistake  has  been  made  in  interpreting  or  treating 
the  observations. 

The  Oxford  values  of  c  are  unusually  discordant.  The  pre- 
sumption that  this  discordance  arises  mainly  from  the  special 
personal  equation  in  observations  of  the  Sun,  described  on 
page  17,  derives  additional  weight  from  the  greater  relative 
consistency  of  the  values  of  6e"  and  e"dn".  I  have  therefore 
allowed  the  values  of  these  quantities  to  receive  a  fair  weight. 

The  value  of  c  for  Paris,  1866-'70,  has  received  a  much  re- 
duced weight,  solely  on  account  of  its  excessive  value.  It 
seems  that  the  work  of  one  observer  who  made  many  observa- 
tions during  this  period  was  affected  by  an  unusual  system- 
atic error. 

Results  of  observations  of  the  Sun's  Right  Ascension. 

GREENWICH. 


Years. 

T 

P' 

w 

c 

W 

6e" 

*"<Jir" 

za 

!750-'62 

-•94 

—.027 

20 

+o'-33 

i-S 

a 
+0.04 

—  0.42 

2 

I765-'7I 

—.82 

—.041 

10 

+0-37 

o-5 

—0.08 

—  o.  64 

I 

1772-^8 

-•75 

—  .022 

10 

+0.74 

o-5 

—  o.  16 

—0.49 

I 

'779-'85 

-.68 

—  -°35 

IJ 

+2.89 

0.  2 

—o.  1  8 

o.  73 

°-5 

ij86-'()2 

-.61 

—•037 

8 

+  i-5i 

0.  2 

—  0.  12 

—0.88 

o 

i793-'97 

-•55 

—  .  114 

5 

+  1.87 

0.  2 

—  O.  22 

—  1.27 

0 

i798-'o2 

—•So 

+.  060 

5 

+  i.  02 

O.  2 

—  o.  42 

—  i.  15 

0 

i8o3-'o6 

—•45 

—  .  002 

5 

+0.27 

O.  2 

—  o.  03 

—1.03 

o 

1807-'  10 

—.41 

—  .068 

5 

—0.34 

O.  2 

-0.32 

—  I.  12 

o 

i8ii_'i4 

—•37 

—•095 

3 

-3-33 

0.  2 

+0.17 

-I    08 

o 

i8i5-'i8 

-•33 

-.052 

6 

—  1.99 

o-5 

—0.  12 

—0-34 

0 

l8l9-'22 

—.29 

+  .010 

6 

—0.51 

+0.  22 

—o.  19 

I 

1823-^6 

—•25 

—•054 

6 

—i.  08 

+0.05 

—o.  17 

I 

i827-'30 

—  .  21 

—.045 

6 

—0.42 

—  o.  09 

—0-75 

I 

i83i-'34 

—  •  17 

+.016 

7 

+0.76 

+0.04 

—o.  27 

I 

1835-38 

—  •  !3 

+  .  O2O 

8 

4*1.  16 

+0.26 

+0.06 

2 

1839-42 

—.09 

+  .061 

8 

+0.84 

+0.32 

+o.  10 

2 

1843-46 

—•05 

-.008 

8 

+0.15 

2 

+0.25 

+0.  22 

2 

i847-'5o 

—  .01 

—  .045 

8 

—  0.  10 

2 

+0.  28 

+0.02 

3 

i8si-'54 

+.03 

+.024 

8 

+0.40 

3 

+O.  22 

-j-o.  02 

3 

1855-58 

+.07 

—.032 

9 

+0.36 

3 

+0.15 

+O.  O2 

3 

i859-'62 

+.11 

—.043 

—0.02 

3 

+  0.25 

+0.  22 

4 

i863-'66 

+.15 

—  .Ol6 

8 

+0-31 

3 

+0.23 

—0.05 

4 

i867-'7o 

+.19 

+  .031 

8 

+Q-35 

3 

+0-33 

—  O.  IO 

4 

i87i-'74 

+.23 

+  .021 

8 

+0.  12 

3 

+0.24 

+q.os 

4 

1875-78 

+.27 

—.008 

8 

.  —  0.  12 

3 

+0.26 

+0.06 

4 

i879-'82 

+•31 

+.017 

8 

—  o.  05 

3 

+0.  21 

+o.  14 

4 

i883-'88 

+.36 

+  .  OOI 

13 

—0.  20 

3 

+0.18 

+0.07 

4 

i889-'92 

+.41 

—.025 

8 

—  0.44 

2 

+o.  24 

+0.  II 

3 

12J  SOLUTION  OF  THE  EQUATIONS.  23 

Results  of  observations  of  the  Sun's  Right  Ascension — Continued. 

PARIS. 


Years. 

T 

/"' 

w 

c 

w 

6e" 

e"fa" 

W 

iSoi-'oy 

-.46 

—  .025 

H 

_i'/78 

0-5 

+o'.'o8 

n 
—0.23 

1  808-'  1  5 

-38 

+.015 

17 

-0.65 

o-5 

—  O.  OI 

+O.  12 

1816-22 

-•3i 

—  .  050 

H 

+o.  18 

0-5 

—o.  13 

+0.32 

i823-'29 

—.24 

—.050 

10 

-j-O.  01 

0-5 

—0.31 

—  O.  O2 

i837-'44 

—.09 

-.034 

19 

+Q-33 

i 

—  o.  04 

+O.  IO 

•5 

i845-'52 

+  .01 

-f-.  009 

15 

+0.  10 

+0.04 

+o.  10 

•5 

i853-'59 

+.06 

+.014 

15 

+0.66 

—  o.  04 

+0.32 

2 

i86o-'65 

+-I3 

+.003 

10 

+0.38 

+0.07 

+o.  26 

2 

i866-'7o 

+.  18 

.000 

7 

+2.29 

.  3 

+°-I3 

+0.40 

2 

i87i-'79 

+  •25 

+.048 

ii 

—o.  26 

—  o.  06 

+O.  22 

2 

1  880-'  89 

+  •35 

-[-.  002 

14 

+  0.44 

+0.24 

-i-o.  03 

2 

PALERMO. 


i79i-'96 

-.56 

—.079 

0 

—  o.  07 

o 

—  o.  06 

—  o'.'ss 

0 

i797-'oi 

-•51 

—  .  116 

0 

—2.33 

o 

—  o.  29 

—o.  28 

0 

i8o2-'o5 

-.46 

—  .  OOI 

0 

—3-" 

o 

-0.05 

—o.  76 

0 

1  806-'  1  2 

—.41 

+•243 

o 

4-5-92 

o 

—1.17 

+  1-55 

o 

CAMBRIDGE. 


1  828-'  34 

.21 

+  .007 

16 

—  o'.'i3 

2 

+0.08 

+0.12 

4 

i835-'40 

—  .  12 

-•033 

H 

—o.  1  8 

2 

+0.06 

—0.  06 

4 

i8|2-'47 

-•05 

—  .  026 

9 

—  O.  21 

2 

+  0.08 

—  O.  12 

4 

i8so-'58 

+  .04 

—  .  O24 

20 

—0.  II 

2 

+0.17 

—0.04 

4 

WASHINGTON. 


1  846-'  5  2 

—  .  01 

-.038 

5 

-o."85 

2 

+0.  20 

o.  oo 

3 

i86i-'6s 

+-I3 

-.038 

8 

-0.53 

4 

+  O   OI 

o.  oo 

5 

i866-'73 

+.20 

—  .004 

13 

—0.  22 

4 

+o.  18 

—  o.  03 

6 

i874-'8i 

+  .28 

-•033 

12 

-0.45 

4 

+0.07 

—  o.  16 

5 

i882-'9i 

+  •37 

—  .  002 

17 

—0.79 

4 

+0.07 

—  o.  07 

5 

KONIGSBERG. 


II 

i8i6-'23 

—•3° 

+  .002 

'3 

+0.30 

I 

+0.07 

—0.28 

3 

1824-^0 

—•23 

—  .006 

12 

+O.  O2 

I 

—  o.  1  6 

+  O.  II 

3 

i83i-'38 

—  -  15 

—  .  O2I 

15 

+0.23 

I 

—  O.  12 

+0.03 

3 

1  839-45 

—.08 

—  .  O2  1 

12 

+0.77 

1 

+0.08 

+o.  20 

3 

24  OBSERVATIONS   OF  THE   SUN.  [13 

Results  of  observations  of  the  Sun's  Right  Ascension — Continued. 

OXFORD. 


! 

j 

Years. 

T             S 

w 

e 

w 

•fe" 

e"W 

w 

i840-'49 

—•05 

—.043 

12 

a 
+2.49 

°-3 

+0.24 

—0/17 

2 

i86o-'68 

+.14 

+.042 

13 

+  1.96 

o-3 

4-0.08 

—o.  13 

2 

1  869-'  76 

+.23 

+  •054 

IS 

4-0.  92 

0-3 

4-0.  20 

—  o.  04 

2 

i88o-'87 

+.34 

-.014 

9 

—0.31 

o-3 

4-0.27 

4-0.64 

2 

PULKOWA. 


1  842-'  50 

i86i-'7o 

—.04 
4-.i6 

4-.  047 
4-.  002 

II 

10 

+  i!'a© 
—  0.40 

i 

—0.  12 

4-0.05 

4-0.  20 
4-0.28 

3 

3 

DORPAT. 


I823-'30 

i83i-'38 

-•23 
-•15 

4-.  021 

+  .008 

I 

4-0.  36 
+0.45 

! 

—0.  12 

4-0.02 

ii 
—  O.  22 
+0.03 

2 
2 

CAPE  OF  GOOD  HOPE. 


1  884-'  90 

+•37 

—  .  026 

12 

—  o.  36 

3 

-f-o!  02 

+0.  OI 

4 

STRASSBURG. 


i883-'88 

+.36 

—.014 

12 

-t!'«s 

2 

4-0.  23 

-fO/09 

3 

The  mass  of  Venus. 

13.  The  mean  results  for  the  mass  of  Venus  given  by  the 
work  at  the  several  observatories  are  shown  as  follows: 

The  probable  error,  where  given  at  all,  is  that  derived  from 
the  discordance  of  the  separate  individual  results  at  the  par- 
ticular observatory.  In  some  cases  there  are  only  one  or  two 
results;  here  no  probable  error  could  be  assigned. 

w'  is  the  sum  of  the  weights  of  the  result  at  each  separate 
observatory,  as  given  by  the  equations  of  condition.  Were 
all  the  observations  of  equal  accuracy,  these  would  be  the 
weights  to  be  assigned  to  the  separate  results.  Such  not  be- 


14] 


CORRECTIONS  OF  RELATIVE  RIGHT  ASCENSIONS. 


25 


ing  the  case,  we  choose  for  the  actual  weights  certain  numbers, 
founded  partly  on  a  compromise  between  the  mean  errors  fol- 
lowing each  result  or  upon  the  values  of  iv1,  partly  on  a  judg- 
ment of  the  accuracy  of  the  observations. 

Values  of  //'  for  the  mass  of  Venus. 


/ 

w 

W 

Greenwich  

—  .  OI5-J-.  006 

226 

II 

Paris  ._.      _            _     __      _ 

—  .  OO7-4-.  OOQ 

146 

c 

Konigsberg  ._   

—  .  OI2-J-.  OIO 

C2 

7 

Cambridge  

—  .  018^.  009 

Co 

6 

Dorpat   _ 

+.  016 

1C 

i 

Pulkowa  

+-025 

21 

i 

Oxford        _   .              . 

4-.  oiA-L.  023 

49 

i 

^^ashington 

—  .  oi84-.  OOQ 

cc 

4 

Cape 

—  .  026 

12 

I 

Strassburg  _         

—  .  014 

12 

I 

Using  the  weights  in  the  last  column,  we  have  for  the  mean 
result 

fi  =  -  .0118  ±  .0034. 

The  mean  error  i  .0034  is  that  given  by  the  discordance  of 
the  separate  results  of  the  preceding  table. 

Corrections  of  relative  Right  Ascensions. 

14.  The  true  values  of  the  remaining  quantities  c,  6e",  and 
e"Sn"  are  to  be  regarded  as  increasing  uniformly  with  the 
time  and  therefore  of  the  form 

x+Ty. 

Here  T  is  the  time,  and  in  the  treatment  of  these  particular 
equations  it  is  counted  from  1850  in  units  of  one  century,  so 
that  x  is  the  value  of  the  correction  at  this  mean  epoch. 

The  quantity  designated  by  c  is  the  same  which,  elsewhere 
in  this  discussion,  is  represented  by  dl  +  a,  so  that 

c  =  dl"  -f  a 

I  shall,  however,  for  convenience,  continue  to  use  the  designa- 
tion c,  or  #+T  y. 


26  OBSERVATIONS  OF  THE   SUN.  [14 

As  the  observations  at  Greenwich  and  Paris  extend  over 
longer  periods  than  at  any  other  observatories,  I  shall  first 
solve  them  separately.  The  totality  of  the  Greenwich  obser- 
vations give  for  c  the  following  normal  equations  and  solution  : 

43.4  x  +  1.65  y=  +  4".23 
1.65  4-  4.24     =  -  I".  25 

x  =  4  0".ll 
y  =  -  0".34 

Those  at  Paris  give  the  equations  and  solution 

8.3  x  +  0.04  y=  +  1".22 
0.04  4-  0.48    ==  4-  0".77 

x  =  4-  0".14 
y  =  4  1".  59 

If  we  combine  all  the  other  results  into  a  single  set  of  normal 
equations,  we  have 

40.2  #  +  4.26  T/  =  -10".84 
4.26  4-  2.20    =  -    3".  98 

x=-  0".10 


It  will  be  seen  that  the  results  for  t/,  the  secular  motion,  are 
markedly  discordant.  Indeed,  if  we  refer  to  the  exhibit  of 
results,  p.  23,  we  shall  see  that  the  values  of  c  are  much  more 
discordant  than  those  of  the  other  two  quantities.  To  obtain 
a  definite  value,  founded  on  all  the  observations  of  the  Sun's 
Eight  Ascension,  I  do  not  see  that  any  better  result  can  be 
obtained  than  that  found  from  a  general  solution  of  the  com- 
bined normal  equations.  The  equations  and  their  solution  are 
as  follows  : 


91.9  #+  5.95  #  =  -5".39 
5.95  +  6.92    =  -  4".46 

x  =  -  0".02 
y  —  —  0".63 
or 

61"  4-  a  =  -  0".02  -  0".63T 


15]      CORK.  TO  THE  SOLAR  ECCENTRICITY  AND  PERIGEE.       27 

Corrections  to  the  solar  eccentricity  and  perigee. 

15.  1  have  already  mentioned  the  remarkable  consistency 
of  the  corrections  to  these  elements  given  by  the  results  at 
different  observatories  and  at  different  epochs.  The  eccen- 
tricity is  more  consistent  than  the  perigee.  One  cause  for 
this,  the  consideration  of  which  will  throw  some  light  on  the 
relative  merits  of  the  observations,  is  that  the  error  of  Bight 
Ascension  depending  on  the  Declination  of  the  object  observed 
effects  the  eccentricity  less  than  the  perigee.  It  is  well  known, 
from  a  comparison  of  the  results,  that  the  systematic  differ- 
ences in  the  Eight  Ascensions  of  different  star  catalogues 
vary  somewhat  with  the  Declination.  Now,  since  the  Sun's 
Declination  goes  through  an  annual  period,  it  follows  that  this 
error  will  produce  a  systematic  effect  on  both  the  eccentricity 
and  the  perigee.  But  the  effect  will  be  much  larger  in  the 
case  of  the  latter  element  than  in  the  case  of  the  former, 
because  of  the  nearness  of  the  perigee  to  the  winter  solstice, 
the  difference  being  only  some  10°  or  12°.  Consequently  the 
extreme  coefficients  in  the  correction  to  the  eccentricity  have 
nearly  the  same  values,  with  opposite  signs,  for  the  same  Decli- 
nations in  different  seasons  of  the  year.  But  it  is  different 
with  the  perigee.  The  coefficient  of  this  quantity  is  negative 
from  October  until  March,  when  the  Sun  is  in  south  Declina- 
tion, attaining  its  maximum  value  about  January  1;  while  it 
is  positive  during  the  remaining  months  when  the  Sun's  Decli- 
nation is  north,  attaining  its  maximum  value  about  July  1. 
A  systematic  difference  in  the  errors  of  Eight  Ascension  will 
therefore  produce  its  full  effect  on  the  longitude  of  the  perigee, 
while  its  effect  on  the  eccentricity  will  be  but  slight. 

In  this  connection,  the  very  large  negative  values  of  the  cor- 
rection to  the  perigee  during  the  period  when  the  old  Green- 
wich transit  instrument  was  in  use  are  quite  remarkable. 
The  progressive  change  in  the  value  of  c  is  also  remarkable  in 
this  connection.  It  is  to  be  remarked  that  the  new  transit  was 
mounted  in  1816,  but  account  was  not  taken  of  this  fact  in 
grouping  the  equations.  Hence  it  is  only  from  the  year  1819 
that  the  results  of  the  table  are  derived  wholly  from  observa- 
tions with  the  new  instrument.  The  anomaly  alluded  to  is 


28  OBSERVATIONS   OF   THE   SUN.  [15 

then  seen  to  disappear.  The  fact  that  the  abnormally  large 
corrections  in  c  are  positive  before  1800  and  negative  after  it, 
while  e"  dn"  is  abnormally  negative  through  the  doubtful 
period  1765-1815,  complicates  the  theory  of  these  errors.  I 
have  not  been  able  to  consider  them  in  detail,  but  have  simply 
rejected  the  results  for  de"  and  e"  6n"  from  1786  to  1818,  hav- 
ing given  them  a  gradually  diminishing  weight  from  BRAD- 
LEY'S  observations  to  the  first  epoch. 

As  in  the  case  of  c,  I  have  made  a  solution  for  Greenwich 
alone,  Paris  alone,  the  other  observatories  combined,  and  all 
combined.  The  results  are  shown  as  follows  : 

1.  From  Greenwich  observations  : 

8e"  e"8Tt" 

54.5#  +  2.73s/  =  +  HM4;  -  0".88 

2.73  +  5.72  =  -}-  1".82$  +  2".69 
x=  +  0".19;  -0".04 
y=  +  0".22  ;  +  0".49 

2.  From  Paris  observations  : 

de"  e"dn" 

17.0#+  0.39  y=  +  0".30;  +  2".95  ' 

0.39  +  0.99  =  +  0".29;  +  0".33 
x=  +  0//.01;  +OM7 
y=  +  0^.29  ;  +  0".27 

3.  The  equations  and  results  from  all  the   other  modern 
observations  are  — 

de"          e"8n" 
77.0#+  4.99#  =  +  5".  58;  +  0".35 

4.99  +  3.68    =  +  1".09;  +  0".40 


y=  +0".22; 


16]      RESULTS  OF  OBSERVED  DECLINATIONS  OF  THE  SUN.      29 

4.  Finally,  if  we  combine  all  the  equations,  we  have  — 

de"  e"d7t" 

US.Zx  -f    8.11  y  =  +  17".02;  +  2"A2 

8.1    -4-10.39     =+    3".  20-,  +  3".42 

a?  =4.    0".10;       0".00 

y=+    0".23;  +  0".33 

In  the  case  of  the  eccentricity  the  general  accordance  is 
quite  satisfactory,  and  for  the  perigee  it  is  much  better  than 
in  the  case  c,  the  relative  Eight  Ascension. 

Results  of  observed  declinations  of  the  Sun. 

16.  The  Sun's  absolute  longitude  can  be  found  only  from 
observations  of  his  declination,  because  this  longitude  is 
referred  to  the  equinox,  which  is  defined  only  by  the  Sun's 
crossing  of  the  equator. 

The  corrections  to  the  eccentricity  and  perigee,  as  just  found, 
are  so  slight  that  they  may  be  neglected  in  determining  the 
correction  of  the  absolute  longitude  from  that  of  the  declina- 
tion. Thus,  as  already  stated,  the  unknown  quantities  of  the 
equations  given  by  the  declinations  are  the  corrections  of  the 
mean  longitude  Z",  and  of  the  obliquity  f,  and  a  constant  A6^ 
peculiar  to  each  observatory,  of  which  we  take  no  further 
account.  The  equation  of  condition  given  by  each  observa- 
tion or  group  of  observations  is 


Ad  4-  A  sin  s  61"  +  Bfo  =  dd 

where  dd  is  the  excess  of  the  observed  over  the  tabular  decli 
nation,  and 

A  =  cosec  e       =  cos  a 


B 


30 


OBSERVATIONS   OF  THE   SUN. 


[16 


The  equations  are  grouped  and  solved  for  periods,  as  in  the 
case  of  the  Eight  Ascensions,  with  the  results  shown  in  the 
following  table: 

Results  of  observations  of  the  Sun's  Declination. 

GREENWICH. 


Years. 

T 

<?/" 

w 

6. 

IV 

* 

,. 

w 

1753-57 
i758-'62 

-•95 
—.90 

+0.78 
+I-5° 

—0-34 

—  1.81 

I 
I 

—2.43 
—1.94 

—0-34 
—  1.81 

I 

I 

1765-^0 

—.82 

—0.23 

o.  95 

0-5 

+o.  20 

—0-95 

o.  5 

i77i-'78 

—  .  75 

+0.48 

—0-93 

0-5 

+  i.  25 

—0-93 

o.  5 

i779-'85 

—  .  68 

+  i.  23 

—  1.09 

—0.99 

—1.09 

o.  5 

i786-'9i 

—.61 

+0.48 

—  o.  50 

o.  3 

f  o.  15 

—  o.  50 

0-3 

1792—  '97 

-•55 

+  1.  12 

—  o.  70 

0.  2 

—0.35 

—  o.  70 

0.  2 

1798-03 

—•49 

+  0.41 

—  1.02 

0.  I 

—  0.  10 

—  1.02 

O.  I 

1  804-'  i  o 

—•43 

+O.I8 

—  I.4I 

O.  I 

—0.84 

—  I.4I 

0.  I 

i8i2-'i6 

—  36 

-0.15 

3 

—0-53 

3 

+0.48 

—0-53 

3 

l8l7-'22 

—•30 

—  o.  41 

3 

+o.  03 

3 

+0.40 

+0.03 

3 

i823~'28 

-.24 

+0-43 

3 

—  0.  10 

3 

+0.08 

—  0.  10 

3 

1  829-  '34 

—.18 

—0.08 

3 

+0.  21 

3 

4-0.25 

+0.21 

3 

1835—  '40 

—  .  12 

—  0.  12 

3 

—  O.  2O 

3 

4-0.37 

—o.  13 

3 

1  84  1  -'46 

—  .   6 

+0.  21 

3 

4-0.13 

3 

4-0.47 

+0.  12 

4 

1847-52 

o 

4-0.25 

4 

O.  OO 

4 

—  o.  24 

—o.  15 

4 

1853-58 

+.    6 

-f°'55 

5 

+o.  18 

5 

—  o.  26 

—  0.05 

5 

1859-64 

+  .12 

+0.03 

5 

+0.28 

5 

—  0.46 

+0.  12 

5 

i865-'7o 

+.18 

-0.23 

5 

—0.15 

5 

4-0.05 

—  0.36 

5 

i87i-'76 

+.24 

—o.  15 

5 

+  0.26 

5 

+o.  16 

—o.  1  6 

5 

i877-'82 

+.30 

—  o.  90 

5 

+0.22 

5 

4-0-34 

+0.08 

5 

1  883-'  88 

+.36 

—0.27 

5 

4-0-33 

5 

-0.14 

-j-o.  02 

5 

i889-*92 

+.4-1 

—0.05 

3 

4-0.19 

3 

4-o.  13 

—  o.  07 

3 

1 

PARIS. 


i8oo-'o3 

—.48 

-f-o.  01 

I.  93 

—  O.  45 

i8o4-'o7 

—  .44 

4-0.7° 

+0.82 

—2.02 

1  808-'  10 

—.41 

+2.66 

+  1.60 

—  °.  95 

i8ii-'i5 

—  .  37 

—  o.  92 

—  I.  2O 

—i.  18 

i8i6-'2i 

—  •  31 

4-0.58 

+  1.68 

—  i.  42 

l822-'28 

—  .  25 

+  1.09 

7 

+o.  39 

7 

—  O.  OI 

i837-'42 

—  .  10 

+o.  79 

7 

—o.  15 

3 

+0.40 

1843  '48 

—  •  4 

+o.  43 

•2 

—  o.  03 

3 

-f-o.  iq 

1849-^4 

-f.  2 

+  i.  19 

2 

—  O.  OI 

2 

+  1.  74 

i855-'6o 

-f.  8 

+o.  35 

o.  02 

7 

+  1.  22 

1  86  1  -'66 

4-.  H 

+  1.35 

3 

o.  oo 

T. 

+O.  12 

i867-'72 

-f-.20 

+o.  31 

2 

—  o.  67 

2 

+O.  IO 

i873~'77 

-f.  25 

O.  Sq 

2 

+o.  04 

2 

•4-1.  01 

i878-'83 

4--31 

—  o.  oq 

2 

—  o.  32 

2 

+o.  q8 

1  884-'  89 

4-.  37 

—0.80 

2 

+o.  32 

2 

+o.  78 

16]     RESULTS  OF  OBSERVED  DECLINATIONS  OF  THE  SUN.       31 

Results  of  observations  of  the  Sun's  Declination — Continued. 

PALERMO. 


Years. 

T 

w 

w 

6e 

W 

J<? 

Vt 

W 

r  -j 

// 
I    46 

o 

// 

O    Qi» 

// 
4-o.  78 

// 

—  o.  o<> 

o  4 

i/y1    UJ 
i  SOA  '  1  1 

-  JJ 
41 

1    I    7O 

o 

O    $2 

4-O.  42 

—  O.  52 

O.  A 

1  0^4     i  j 

CAMBRIDGE. 


1877  '78 

T  A 

O  "I 

2 

O.  77 

4-o.  So 

—  O.  54 

i 

10OJ   O° 

1870—  '44 

—  .08 

-[-o.  31 

2 

—  O.  2O 

+o.  29 

—  o.  41 

i 

1847—  '57 

oo 

4-O.  21 

2 

4-o.  7,1 

—  O.  }2 

-fo.  10 

i 

i854-'s8 

+.06 

—  o.  15 

2 

4-o.  74 

—  o.  42 

-4-o.  13 

i 

WASHINGTON. 


i840-'49 

—  .  02 

—0.28 

4 

—o.  73 

—o.  47 

—  o.  81 

2 

i86i-'66 

+.14 

—  O.  II 

4 

—  o.  43 

—0.45 

—  o.  25 

2 

i867-'72 

+  .  20 

4/-O.  74 

4 

—  o.  39 

+0.28 

—  o.  51 

2 

i8y3-'78 

+  .  26 

—o.  58 

4 

—  o.  32 

-(-o.  10 

—  o.  45 

2 

1  879-'  84 

+  •32 

—  o.  31 

4 

—  o.  60 

—0.35 

—  o.  72 

2 

i88s-'9i 

-f-38 

—  o.  02 

4 

—  0.05 

—  o.  20 

—o.  18 

2 

KONIGSBERG. 


i8i 

I  O7 

1  820-'  23 

—  28 

—  o.  14 

2 

—  O.  22 

—  °-  59 

—  O.  47 

I 

—  .  24 

-(-0.65 

2 

4~°-  49 

—  o.  60 

-f-o.  24 

I 

i828-'3i 

—  .  20 

41.08 

2 

+o.  09 

—  o.  64 

—  o.  1  6 

I 

1  832-'  34 

—  .  17 

—  o.  72 

2 

—  o.  15 

—  I.  72 

—  o.  40 

I 

1  837-'  44 

—  .  09 

—0.66 

2 

—  o.  62 

—  2.  24 

—  o.  87 

I 

OXFORD. 


1  840-'  45 
1  846-'  5  1 
i86i-'66 

—.07 
—  .  01 

+•  H 

+0-79 

+0.35 
-j-o.  36 

2 
2 
2 

+0.42 
4-0-40 

—  o.  81 



+0.67 
4-0.89 

4-o.  10 

4-0.  22 
4-0.  20 
—  I.  OI 

O.  2 
0.  2 

O  2 

1  867-'  7  2 

+.20 

—  o.  1  6 

2 

—  o.  24 

-j-O.  2Q 

—  O.  44 

O  2 

1873-76 
i88o-'83 

+  .25 
+  •32 

—0.38 
—  °-43 

2 
2 

—0-33 

-f-O.  12 



+0.29 

—  o.  17 

—o  53 
—o  08 

0.  2 
O.  2 

i884-'87 

-f-36 

—  o.  24 

2 

-fo.  23 

—  o.  19 

4-o.  07 

O.  2 

OBSERVATIONS   OF  THE   SUN.  [16,  17 

Results  of  observations  of  the  Sun's  Declination — Continued. 

PULKOWA. 


Years. 


61" 


W  6e 


\\ 


W 


1842^45       —.06       -fO.  82               2       —0.35     — 0.01  —0.35                  I 

—.02     — o.  10           2     —0.48    +0.07  — 0.48          .1 

i86i-'65     +.13     —0.53           2     —0.48    —0.30  —0.48 

i866-'7o     -f.  18     +0.27           2     —0.31    —0.38  —0.31 

DORPAT. 

i823-'28     —.24     +0.99           2     —1.26    +0.59  —1.41             i 

i829-'32     —.19     -f°- 99           2     — o.  76    -fI-34  —0.91             I 

l833~'38     —.14     4-1-00           2     —0.63    -fi-34  —0.78            i 

CAPE  OF  GOOD  HOPE. 

i884-'87     +.36     — 0.51           4     +0.05    +o.  ii  —0.07             2 

i888-'9O     -}-•  39     — 0.84           4     -{-0.09    -f0-  19  — 0.21             2 

.  STRASBURG. 

i884-'88     +.36     —0.57           4     —0.05    —0.77  4-0.12             2 

LEIDEN. 

i864-'69     -)-.  17     +0.14           4     — o.  01    -(-0.27  — 0.24             2 

i87o-'76     -(-.23     — 0.23           4     — 0.06    — 0.04  — 0.29             2 


Correction  to  the  Sun's  absolute  longitude 

17.  So  far  as  mere  instrumental  measurement  is  concerned, 
the  correction  d  s  should  be  determined  with  greater  precision 
than  dl"  in  the  ratio  5:2,  because  the  errors  in  decimation 
have  to  be  divided  by  the  factor  sin  s  =  0.40,  in  order  to  form 
dl".  Allowing  for  this  large  increase  in  the  source  of  error, 
the  values  of  6 1"  are  more  accordant  than  those  of  6  8.  This 
is  what  we  should  expect.  The  values  of  the  former  quantity 
depend  mainly  upon  the  comparison  of  observations  made 


17,  18]  OBLIQUITY   OF  ECLIPTIC.  33 

near  the  opposite  equinoxes,  when  the  snn  has  the  same  decli- 
nation, and  when  the  season  is  not  greatly  different.  Indeed, 
if  the  season  changed  exactly  with  the  sun's  declination,  all 
effects  of  annual  change  of  temperature  would  be  completely 
eliminated  from  61",  as  would  also  in  any  case  any  constant 
error  which  is  a  function  simply  of  the  Sun's  Declination.  It 
is  therefore  to  be  expected  that  the  actual  probable  error  of 
this  quantity  will  conform  more  nearly  to  that  determined  from 
the  residuals  than  in  the  case  of  the  other. 

For  these  reasons  the  value  of  dl"  does  not  give  rise  to 
much  discussion.  The  general  result  from  all  the  observa- 
tories is,  for  dl",  when  developed  in  the  form  x  -f-  y  T. 

x  =  +  0".05 
y  =  —  0".97. 

Obliquity  of  the  ecliptic. 

18.  The  determination  of  the  obliquity  rests  upon  an  essen- 
tially different  basis  from  that  of  the  absolute  longitude,  in 
that  it  depends  upon  actual  differences  of  measured  Declina- 
tions, which  differences  are  still  further  complicated  by  the 
fact  that  they  are  necessarily  made  at  opposite  seasons.  A 
more  detailed  discussion  of  them  is  therefore  necessary,  and 
some  modification  may  have  to  be  made  in  the  separate  results 
as  adopted.  The  following  special  circumstances  affecting  the 
observations  are  to  be  taken  into  consideration : 

The  BRADLEY  Greenwich  results  for  1753-^2,  are  derived 
from  a  manuscript  communicated  by  Dr.  AUWERS,  containing 
the  results  of  his  very  careful  reduction  of  BRADLEY'S  ob- 
served Declinations  of  the  Sun,  which  were  compared  with 
HANSEN'S  tables.  The  corrections  were  reduced  to  those  of 
LEVERRIER'S  tables  by  being  computed  at  intervals  suffi- 
ciently short  to  permit  of  the  reduction  being  interpolated  with 
all  necessary  precision.  No  reduction  was  applied  either  on 
account  of  the  constant  error  of  the  Declinations  determined 
by  Dr.  AUWERS  himself,  nor  for  reduction  to  the  Boss  system 
of  standard  Declinations.  Hence  arises  the  large  value  of  Ad 
given  by  these  Declinations.  Consequently  the  value  of  df  is 
5690  N  ALM 3 


34  OBSERVATIONS  OF  THE  SUN.  [18 

that  given  immediately  by  the  instrument,  on  the  system  of 
reduction  adopted  by  Dr.  AUWERS,  in  which  I  have  supposed 
that  the  Pulkowa  refractions  were  used. 

From  17G5  to  1816  the  Greenwich  observations  were  made 
with  the  imperfect  quadrant,  the  Declinations  of  which  are 
subjected  to  an  error  which  is  not  constant.  The  neces- 
sary corrections  are  derived  by  S AFFORD  in  Vol.  n  of  the 
Astronomical  Papers.  The  corrections  are  those  necessary  to 
reduce  to  Boss's  system,  and  they  vary  with  the  Declination. 
Hence  the  arc  on  which  the  obliquity  depends  is  not  that 
measured  with  the  instrument  itself,  but  that  so  corrected  as 
to  reproduce  as  nearly  as  may  be  the  standard  Declinations. 

From  1812  onward  the  two  mural  circles  were  used.  Up  to 
1830  no  correction  except  the  constant  one  derived  by  SAF- 
FORD  was  applied  to  the  Declinations  as  measured  with  these 
instruments.  Hence  the  arc  of  obliquity  is  that  measured 
with  the  instrument  itself  without  being  corrected  by  the 
standard  stars. 

After  1830  the  Declinations  were  corrected  by  the  tables  for 
Greenwich  given  in  Boss's  paper.  These  corrections  vary 
somewhat  with  the  Declination,  and  they  are  different  also 
for  different  periods.  Hence  we  have  here  a  period  during 
which  the  instrumental  differences  of  Declination  were  cor- 
rected to  reduce  them  to  the  standard  star- system. 

If  the  standard  system  were  subject  to  no  farther  error  than 
a  constant  one,  common  to  all  Declinations  within  the  zodiac, 
which  common  correction  would  be  subject  to  a  uniform  change 
with  the  time,  this  system  would  doubtless  be  the  best  one  to 
adopt  in  order  to  obtain  the  secular  variation  in  the  obliquity 
of  the  ecliptic.  But,  as  a  matter  of  fact,  the  standard  Decli- 
nations are  simply  the  mean  results  of  Declinations  measured 
with  different  instruments.  It  is,  therefore,  a  question  whether 
we  shall  get  any  better  results  by  applying  reductions  to  a 
standard  system  than  we  should  get  by  simply  taking  the 
mean  of  the  instrumental  results,  because  the  system  is  itself 
only  a  mean  of  such  results.  It  is  true  that  the  standard  sys- 
tem depends  on  more  instruments  than  the  obliquity,  though 
not  on  better  ones;  but  it  is  also  to  be  considered  that  the 
reductions  in  the  case  of  the  Sun  may  be  different  from  those 


18,  19]  OBLIQUITY   OF   ECLIPTIC.  35 

in  the  case  of  the  stars,  owing  to  the  very  different  conditions 
in  which  the  observations  are  made. 

Another  troublesome  point  arises  from  the  refraction  used 
in  the  reductions.  The  effect  of  refraction  is  always  to  make 
the  measured  obliquity  less  than  the  actual  one;  the  correc- 
tion to  the  obliquity  on  account  of  refraction  is  therefore  a 
positive  quantity,  which  is  a  minimum  for  an  observatory  at 
the  equator  and  increase  equally  towards  each  pole.  Some 
values  of  the  obliquity  were  derived  from  BESSEL'S  refractions 
of  the  Tabulae  Regiomontance,  and  others  from  the  Pulkowa 
tables.  Since  the  secular  variation  of  the  obliquity  is  more 
important  than  the  absolute  value  of  the  quantity,  it  is  essen- 
tial that  the  standard  to  which  all  determinations  of  the  ob- 
liquity are  reduced  should  be  as  nearly  as  possible  the  same, 
and  therefore  that  the  same  refraction  should  be  used.  But  in 
reductions  to  standard  star  places  we  meet  with  the  addi- 
tional complication  that  the  differences  in  the  constant  of 
refraction  might  be  wholly  or  partially  eliminated  by  the 
reductions  to  a  standard  system.  It  would  therefore  be  a  dif- 
ficult question  how  far  we  should  modify  the  values  of  6s  on 
account  of  the  use  of  different  tables  of  refraction. 

To  avoid  all  these  difficulties  I  have  judged  it  best  to  make 
the  obliquity  depend  mainly  upon  absolute  measures,  the 
reductions  being  made  with  the  Pulkowa  refractions. 

Effect  of  refraction  on  the  obliquity. 

19.  The  determination  of  the  average  or  most  probable  effect 
on  the  obliquity  produced  by  using  the  Pulkowa  refractions, 
instead  of  those  of  the  Tabulce  Regiomontanw,  is  easily  deter- 
mined. We  divide  the  ecliptic  into  a  number  of  equal  arcs 
throughout  the  year,  and  by  equations  of  condition  express 
differences  of  refraction  in  terms  of  differences  of  Declination, 
and  hence  differences  of  obliquity.  We  thus  find  that  at 
certain  latitudes  where  observations  were  made,  and  where 
BESSEL'S  refractions  were  used  in  the  reduction,  the  follow- 
ing corrections  are  necessary  to  reduce  the  obliquity  to  the 
ones  given  by  the  Pulkowa  refractions: 

Pulkowa;         y  =  59°.S;  Jf  =  —  0".325 
Greenwich;      <p  =  51°.o;  z/f  =  —  0".20 
Washington;   q>  =  3S°.9;  4e  =  -  0".125 


36  OBSERVATIONS   OF   THE   SUN.  [19 

Hence  I  conclude  that  for     . 

Dorpat;  As  —  —  0".29 
Konigsberg;  Js  =  —  0/7.26 
Cambridge;  Je  =  -  0".21 
Cape  Town;  At  =  -  0".12 

The  corrections  to  the  obliquity  thus  derived,  depending 
mainly  on  direct  instrumental  measurement,  and  reduced  to  the 
Pulkowa  refractions,  are  designated  as  6'f .  The  results  for  this 
quantity  are  given  in  the  last  column  of  the  several  tables. 

In  the  case  of  BRADLEY'S  Greenwich  results,  I  have  taken 
as  6'e  Dr.  AUWERS'S  results  unchanged,  assuming  in  the 
absence  of  any  specific  statement  that  he  has  used  the  Pul- 
towa  refraction  tables. 

In  the  case  of  MASKYLENE'S  observations,  I  have,  by  excep- 
tion, used  them  as  reduced  to  the  standard  star-system, 
because  we  have  no  other  results  at  these  times,  and  the  en  or 
of  his  instrument  is  so  strongly  shown  that  it  would  not  do  to 
use  the  results  unchanged.  It  will  be  seen,  however,  that 
small  weights  are  assigned,  and  that  the  weights  diminish 
towards  the  end  of  the 'series. 

In  the  case  of  the  Greenwich  observations  from  1812  to 
about  1834,  no  change  has  to  be  made,  as  the  results  are  gen- 
erally or  always  purely  instrumental,  and  Pulkowa  refractions 
are  used  in  SAFFORD'S  work. 

From  1835  onward  I  have  depended  mainly  on  certain  cor- 
rected Greenwich  reductions.  First,  for  tf7£,  I  have  used  the 
results  given  by  Mr.  CHRISTIE  in  his  very  valuable  paper  on 
the  Greenwich  Declinations,  in  M.  E.  A.  S.,  Vol.  XLV,  where 
the  Declinations  from  1836  to  1879  are  reduced  on  a  uniform 
system.  Later,  I  have  adopted  the  corrected  results  given  in 
Appendix  III  to  the  Greenwich  observations  for  1887.  In 
each  case  the  result  has  been  reduced  to  the  Pulkowa  refrac- 
tions. 

The  Paris  results  rest  on  a  different  basis  from  the  others, 
in  that  the  zero  point  of  the  instrument  depends  wholly  upon 
LEVERRIER'S  Declinations  of  the  stars,  and  I  fear  it  was  not 
always  accurately  determined.  Observations  near  the  winter 
solstice  are  mostly  referred  to  one  set  of  stars;  those  near  the 


19J  OBLIQUITY   OF  ECLIPTIC.  37 

summer  to  another  set,  the  error  of  which  may  be  systemat- 
ically different.  Certain  it  is  that  the  results  during  the  early 
years  were  very  discordant.  The  weights  as  given  in  the  table 
are  those  assigned  a  priori,  without  sufficient  reference  to  the 
discordance  of  the  older  results.  I  have  felt  constrained  to 
evade  a  decision  as  to  their  treatment  by  entirely  omitting 
their  results  in  the  final  discussion. 

Iii  the  case  of  some  other  observatories  it  was  difficult  to 
determine  exactly  what  refractions  had  been  used  in  each 
special  case  and  what  reductions  should  be  made.  I  have,  how- 
ever, determined  the  corrections  in  the  best  way  I  was  able. 

A  precise  determination  of  the  secular  change  in  the  ob- 
liquity is  of  more  importance  for  our  present  object  than  a 
precise  determination  of  its  amount.  Hence  a  series  of  obser- 
vations extending  through  a  long  period  of  time,  and  made  on 
a  uniform  system,  has  an  advantage  over  a  number  of  isolated 
values,  in  that  any  constant  error  with  which  it  may  be 
affected  will  be  eliminated  from  the  secular  variation.  Possi- 
ble constant  differences  between  the  determinations  of  the 
various  observatories  at  different  epochs  will  vitiate  the  sec- 
ular variation,  but  the  probable  amount  of  this  error  may  be 
diminished  by  using  a  number  of  separate  determinations, 
such  as  are  presented  in  the  preceding  table.  In  the  Green- 
wich transit  circle  we  have  a  very  uniform  series,  extending 
over  a  period  of  forty  years,  but  giving  results  systematically 
different  from  other  determinations.  This  series  gives  for  the 
correction  to  the  obliquity : 

Transit  Circle,  1847->91 : 

d'e  =  -  0".ll  i  0".06  +  (0".21  i  0".46)  T    .     .     .     (a) 

Here,  in  view  of  the  uniformity  of  method  and  reduction, 
we  may  regard  the  mean  error  of  the  centennial  variation  from 
the  discordance  alone  as  a  fair  approximation  to  the  probable 
mean  error.  It  will  be  seen  that  I  have  here  included  four 
years  (1847-'50)  of  the  Mural  Circle  results. 

Continuing  the  Greenwich  series  backward,  the  question 
arises  whether  we  can  regard  the  results  of  the  mural  circle 
from  1812  to  1850  as  comparable  with  those  of  the  transit  circle. 


38  OBSERVATIONS   OF  THE   SUN.  [19 

There  is  certainly  nothing  in  the  table  to  indicate  any  system- 
atic difference.    From  the  combination  of  the  two  we  have — 

M.  C.  andT.  0.,  1812->50: 

<J'6  =  -  0".08  ±  0".05  +  (+  0".14  i  0".23)  T  (1850)  .  .    (b) 

Here  the  mean  error  is  naturally  smaller  than  in  the  case  of 
the  transit  circle  alone,  but  is  now  more  subject  to  possible 
systematic  difference  between  the  two  instruments. 

If  we  now  go  back  to  BRADLEY,  we  meet  with  the  very  diffi- 
cult question,  whether  we  should  regard  his  results  as  best 
comparable  with  the  modern  Greenwich  observations,  or  with 
modern  observations  in  general.  If  we  assume  that  the  differ- 
ence between  the  Greenwich  and  other  modern  results  is  due 
to  any  cause  which  has  remained  unchanged  since  BRADLEY, 
we  should  reach  one  conclusion;  otherwise,  we  should  reach 
the  other.  The  result  of  combining  all  Greenwich  observa- 
tions, with  the  weights  as  assigned,  is — 

6'e  =  -0".ll  +  0".50T (c) 

In  this  combination  I  have  used  the  weak  results  of  MASKE- 
LYNE,  with  the  small  weights  assigned,  although  they  depend 
wholly  upon  the  standard  declinations  of  stars.  In  view  of 
the  discordance  between  BRADLEY'S  two  results,  this  seems 
the  only  admissible  course. 

Next  in  the  length  of  time  which  they  include  come  the  Paris 
observations,  of  which  the  results,  with  the. weights  assigned, 
are — 

6f=  +  0".01  — 0".36T 

I  give  this  result  in  order  that  nothing  may  be  omitted. 
Undue  weight  has  probably  been  assigned  to  the  earlier 
determinations;  in  any  case  the  method  of  deriving  it  from 
the  original  observations  is  so  objectionable  that  no  further 
use  is  made  of  it.  A  satisfactory  discussion  of  the  observa- 
tions would  require  a  complete  redetermination  of  the  zero 
points  of  the  instrument  from  fundamental  stars. 


19,  20]         DISCUSSION   OF   RESULTS   OF   OBLIQUITY.  39 

If  we  omit  the  Greenwich,  Paris,  and  Palermo  results,  and 
combine  all  the  others  into  a  single  set  of  equations  of  condi- 
tion, we  have  the  equations  arid  results  : 

36.9#  +  0.26  y  =  -  14".37 
0.26   +  1.88    =  +    1".01 

x  =  -  0".39 
y=+  0".59 

Here  x  is  the  value  of  6'e  for  1860,  and  y  its  centennial  varia- 
tion. Transferring  the  epoch  to  1850,  as  usual,  the  result  is  — 

d'e  =  -  0".45  +  0".59  T      .....     (d) 

No  reliable  mean  error  can  be  computed,  owing  to  systematic 
errors.  In  view  of  these,  one  mode  of  treatment  would  be  to 
form  equations  of  condition  in  which  a  possible  systematic 
error  at  each  observatory  would  appear  as  one  of  the  unknown 
quantities.  By  this  .  process  we  should  get  the  same  result 
for  the  secular  variation  as  if  we  made  an  independent  determi- 
nation from  the  work  of  each  observatory.  At  most  of  the 
observatories  the  period  through  which  the  observations  are 
made,  with  one  instrument  and  on  an  unchanged  plan,  is  too 
short  to  render  such  a  course  advisable. 

As  a  last  combination,  we  shall  combine  the  earlier  Green- 
wich results,  up  to  1810,  with  Palermo  and  with  all  the  modern 
results  except  Paris,  first  dividing  the  weights  of  the  Green- 
wich results  by  2.  We  then  have  the  equations  — 

39.8  a?  -1.82  y  =  -  17  ".12 
-  1.8     +  3.47     =  +    2".99 

x  =  —  0".40 

y=+0".65      .......  •      («) 


Concluded  results  for  the  obliquity. 

20.  The  data  on  which  these  various  results  for  the  obliquity 
rest  show  the  following  noteworthy  features  : 

(1)  That  the  correction  given  by  the  modern  Greenwich 
instruments,  mural  and  transit  circles,  is  markedly  greater 


40  OBSERVATIONS   OF  THE   SUN.  [20 

than  that  given  by  other  modern  observations.  This  may  be 
most  plausibly  attributed  to  the  atmospheric  conditions 
within  the  observing  room. 

(2)  The  minuteness  of  the  change  of  the  correction  given 
by  these  instruments  during  nearly  eighty  years.  To  this 
circumstance  is  due  the  smallness  of  the  centennial  variation, 
0".50,  found  from  the  totality  of  the  Greenwich  observations. 
A  comparison  of  BRADLEY  with  the  mean  of  the  T.  C.  results 
only  would  have  given  a  change  of  0".97  in  117  years,  or  a 
centennial  change  of  about  0".80. 

The  long  period,  uniformity  of  plan,  and  systematic  devia- 
tion of  the  modern  Greenwich  observations  lead  me  to  consider 
them  as  forming  a  series  distinct  from  all  others.  We  have 
therefore  the  following  two  completely  independent  determi- 
nations of  the  centennial  variation : 

(1)  Modern  Greenwich  results:   y  =  +  0".14  i  0".23 

(2)  All  other  results  +  0".6o 

To  the  latter  no  reliable  mean  error  can  be  assigned.  To 
judge  its  reliability  we  may  compare  it  with  the  results  («•),  (c), 
and  (d)— 

Greenwich  T.  C.,  alone,  +  0".21  ±  0".46 

Greenwich  observations  in  general,  -f-  0".50 
Miscellaneous  modern  observations,  +  0".59 

We  may,  it  would  seem,  fairly  give  double  weight  to  the 
result  (2),  thus  obtaining,  as  the  definite  result  from  observa- 
tions of  the  Sun  alone: 

Correction  to  LEVERRIER'S  centennial  variation  of  the  obliq- 
uity of  the  ecliptic  (-  47".594) 

+  0".48  ±  0".30 

the  mean  error  being  an  estimate  from  the  general  discordance 
of  the  data. 
For  the  constant  part  of  the  correction  I  take — 

tie  (1850)  =  -  0".30 


21]  SUMMARY  OF  RESULTS.  41 

Summary  and  comparison  of  results. 

21.  From  what  precedes  we  have  the  following  as  the  values 
of  the  unknown  quantities,  and  of  their  secular  variations,  as 
given  by  observations  of  the  Sun  alone. 


de"  = 

Value  for 
1850. 

+  0".10  d 

-  07/.03 

Cent, 
var. 

+  0".23  ± 

0^.10 

e"(dn"+a)  = 

0".00  J 

r  0/7.07 

+  C^.33  i 

0;/.12 

dl"+a  = 

-  0".02 

-  G^.63 

dl"  = 

+  0^.05  J 

-  0".12 

-  0/7.97  i 

0^.23 

ds  = 

-  0/7.30  = 

t  07/.15 

+  077.48  i 

:  0".30 

a  = 

-0".07 

4-  0/7.34 

No  estimate  of  the  probable  errors  of  these  quantities  would 
be  useful  which  did  not  take  account  of  the  systematic  dif- 
ferences between  the  results  of  different  observatories.  We 
have  therefore  formed  the  mean  outstanding  residual  correc- 
tions given  by  the  several  observatories,  as  shown  in  the 
tables  which  follow.  Originally  the  scale  of  weights  used  for 
the  Greenwich  observations  did  not  correspond  to  that  for  the 
other  observatories;  they  were,  therefore,  divided  by  2.  As 
used  below,  however,  the  change  has  been  made  in  the  case 
of  dl"  by  multiplying  all  the  weights  of  the  other  observatories 
by  2,  and,  in  the  case  of  6s,  by  dividing  the  Greenwich  weights 
by  2. 

The  correction  to  the  obliquity  depends  solely  on  6'e  ;  but 
the  comparison  has  also  been  made  with  the  values  of  <?£, 
which,  it  will  be  remarked,  differ  from  the  others  in  that 
account  is  taken  of  the  supposed  variation  of  the  systematic 
correction  with  the  declination.  It  is  noteworthy  that  the 
results  are  somewhat  more  accordant  when  this  correction  is 
omitted  and  purely  instrumental  errors  are  used  for  the 
obliquity. 

The  mean  errors  given  in  the  preceding  summary  of  results 
are  derived  from  the  discordances  in  question,  and  may  be 
regarded  as  substantially  real. 

No  use  was  made  of  the  Paris  results  for  61"  and  ds  for 
the  reason  that  they  depend  on  decimations  referred  to  star 


42  OBSERVATIONS  OF  THE   SUN.  [21 

places  which  may  be  affected  by  differences  in  different  Eight 
Ascensions.  They  are,  however,  retained  in  the  table  to  show 
the  amounts  of  outstanding  discordance. 

Outstanding  mean  residual  corrections  to   quantities  depending 
on  the  Sun's  Right  Ascension. 


Greenwich                    + 

0".09         -  0".03 

54.5 

Paris 

0".09         +  0".17 

17 

Cambridge                    4- 

07/.02             0".00 

16 

Washington 

0".05         —  0".12 

24 

Konigsberg 

0/7.08         4-  0/7.08 

12 

Oxford                   .        4- 

0".06         4-  0".02 

8 

Pulkowa 

0/7.15         4-  077.22 

6 

Dorpat 

0/7.10         -  0/7.03 

4 

Cape 

07/.16         —  07/.ll 

4 

Strassburg                   4- 

077.05         —  (V.03 

3 

Mean  errors  for 

weight  unity    t  \  =  ± 

0/7.34         ±  0/7.39 

Mean  error  of  x            i 

0/7.03         i  0/7.03 

Mean  error  of  y            ± 

0/7.10         ±  0/7.12 

Outstanding  mean  residual  corrections  to  quantities 

depending 

on  the  Sunh 

?  Declination. 

SI" 

w                 de             w 

d'K 

Greenwich                -  07/.06 

64       4-  077.31       29.6 

4-0".!  7 

Paris                        4-  0/7.45 

0       4-  07/.31        0 

Palermo                   -  077.39 

0       -  0/7.20         0.8 

—  0/7.20 

Cambridge               -  0".05 

8       4-  077.35        4 

4-  0/7.14 

Washington            4-  0/7.07 

24       -  07/.22      12 

-  0/7.29  • 

Konigsberg              -  077.20 

10       +  0/7.31        5.5 

0/7.00 

Oxford                      4-  0/7.14 

14       +  0/7.19        1.4 

-  0".01 

Pulkowa                  +  0/7.12 

8       -  077.13        4 

-  0/7.13 

Dorpat                     4-  07/.75 

6       —  077.49        3 

-  0/7.64 

Cape                         -  07/.35 

8       4-  0/7.10        4 

-  0/7.02 

Leiden                     +  0/7.10 

8       4-  077.17        2 

-  0/7.06  . 

Strassburg               -  0/7.26 

4       +  0/7.08        4 

4-  0/7.25 

€  for  weight  unity  ±  0/7.81 

±  07/.74 

±  0/7.60 

CHAPTEE  III. 

RESULTS  OF   OBSERVATIONS  OF    MERCURY,  VENUS,  AND 

MARS. 

Elements  adopted  for  correction. 

22.  We  first  give  an  outline  of  the  method  of  expressing  the 
observed  corrections  to  the  Eight  Ascensions  and  Declinations 
of  each  of  the  planets  as  linear  functions  of  the  corrections  to 
the  tabular  elements.  This  linear  function  forms  the  first 
member  of  the  equation  of  condition  in  its  original  form,  and 
the  observed  correction  forms  its  second  member. 
Let  us  put  — 

E,  r,  the  radii  vectores  of  the  Earth  and  planet  5 
L,  the  Sun's  true  longitude; 

J,  the  inclination  of  the  orbit  of  the  planet  to  a  plane 
passing  through  the  Sun's  center  parallel  to  the 
plane  of  the  Earth's  equator; 
N,  the  Eight  Ascension  of  the  ascending  node  of  the 

orbit  on  this  plane; 

U,  the  argument  of  heliocentric  declination  of  the  planet 
or  its  angular  heliocentric  distance  from  the  node 
on  the  equator; 
a,  6,  the  geocentric  Eight  Ascension  and  Declination  of 

the  planet. 
€,  the  obliquity  of  the  ecliptic; 

We  shall  then  have  — 

a  =/(r.  E.  L.  J.  X.  U.,  *.)    .     .     .     .     .     .     .     .V    (a) 

For  the  correction  to  the  tabular  Eight  Ascension  arising 
from  symbolic  corrections  to  these  seven  quantities,  we  have 
the  equation  — 

Sa  =  A«63  +  *£  tfN  +  %L  SU  +  %  Sr  +  *«  <fe 
dJ  dN  du  dr  de 


43 


44  MERCURY,   VENUS,   AND  MARS.  [22 

with  a  similar  equation  for  the  declination,  formed  from  this  by 
writing  <5  for  a. 

The  relations  by  which  these  two  equations  are  derived,  as 
well  as  the  expressions  for  the  differential  coefficients  they 
contain,  are  given  very  fully  in  A.  P.,  Yol.  II,  Part  I,  to  which 
reference  may  be  made.  The  corrections  tfN  and  tfU  are  not, 
however,  the  most  convenient  ones  to  choose.  It  will  be  found 
in  the  paper  alluded  to  that  they  have  been  transformed  by 
measuring  the  longitude  in  orbit  of  the  planet  and  that  of  the 
perihelion  from  an  arbitrary  point  in  the  orbit.  As  to  this  very 
convenient  device  in  celestial  mechanics,  it  is  to  be  remarked 
that  the  "departure  point"  always  disappears  from  the  final 
equations  which  determine  the  position  of  the  planet.  We 
may,  in  fact,  make  abstraction  of  it  by  considering  that  its 
introduction  is  equivalent  to  the  following  simple  linear  trans- 
formations. 

We  put 

w,  the  distance  from  the  node  to  the  perihelion  ; 

/,  the  true  anomaly  ; 

g,  the  mean  anomaly. 

TT,  the  longitude  of  the  perihelion  ; 

I,  the  mean  longitude  of  the  planet; 

v,  its  true  longitude; 

these  longitudes  being  counted  from  the  departure  point. 
Then,  we  have  the  relations  — 


#U  =  tfw  4-  df  •-•-=  6v  —  cos 
6w  =  dn    -  cos  JtfN  (2) 

61  =  dn    +dg 
Hence, 

dn  =  tfU  -f  cos  JtfN  —  df 

(3) 


The  elements  finally  adopted  for  correction  by  the  equations 
of  condition  were  — 

I.  TT.  e.  J.  N. 


22,  23]          ELEMENTS  ADOPTED  FOR   CORRECTION.  45 

The  value  of  a,  the  mean  distance,  is  known  with  such  pre- 
cision that  its  correction  need  not  enter  into  the  equations  of 
condition.  The  latter  are  formed  by  substituting  in  (1) 


n+-e  +  -  cos 

dgj  de  dg  (4) 

»        dr  x    .   dr  ~,      dr  ^ 

dr  =  -=-  de  +  -=-  61  —  =-  o  n 
de  dg          dg 

The  coefficients  of  each  equation  of  condition  from  the  Eight 
Ascension  thus  become  — 

Coefficient  of  $J     .     .     . 

dS 

™XT  da  da 

»     ...    ^-cosJ^- 

«     de     .     .     .    *£ 

(5> 
u  u      rfi  <*«  ^/  _.    ^  ^r 

ro  ^"^  ^  ^ 

da     .d   \dadr 


u          it     se  dtx  df  4-  da  dr 

Wde  +  W  fo 

In  the  second  members  of  the  equations  a  is  regarded  as 
a  function  of  the  seven  quantities  (a),  as  is  also  #,  for  which 
a  similar  equation  is  to  be  formed. 

The  corrections  of  the  solar  eccentricity,  perihelion,  and 
mean  longitude  were  also  introduced  by  putting  in  (1) 


tfL  =  dl"  +         de"  +       ±  dn" 

de"  dn"  (6) 

^R  =  *®L  de"  +  ^  dn" 

de"  dn" 


Introduction  of  the  masses  of  Venus  and  Mercury. 

23.  The  correction  to  the  mass  of  Venus  was  introduced 
by  taking  the  tabular  perturbation  produced  by  Venus  on 
the  geocentric  place  of  the  planet  at  the  mean  date  of  each 
equation  as  the  coefficient  of  the  unknown  quantity  to  be 
determined.  In  computing  these  perturbations  regard  was 


46  MERCURY,  VENUS,  AND  MARS.         [23,  24 

had  to  the  action  of  Venus  on  the  Earth  as  well  as  ou  the 
planet.  On  this  system  the  unknown  quantity  finally  found 
would  be  the  factor  by  which  the  adopted  mass  of  the  planet 
must  be  multiplied  in  order  to  give  the  correction  of  that  mass. 

It  has  already  been  remarked  that  the  mass  of  a  planet  can 
not  be  determined  free  from  systematic  error  by  observations 
made  upon  the  planet  itself.  Hence,  the  mass  of  Venus  can 
be  determined  only  from  observations  of  Mercury  and  Mars, 
and  that  of  Mercury  only  from  observations  of  Venus  and 
Mars.  But  the  mass  of  Mercury  is  so  minute  that  it  would  be 
useless  to  attempt  to  determine  it  from  observations  either  of 
the  Sun  or  Mars.  It  was  therefore  determined  solely  from  the 
periodic  perturbations  of  Venus. 

It  has  happened  that  the  mass  of  Venus  could  not  be  deter- 
mined in  a  reliable  way  from  observations  of  Mars,  owing  to 
a  defect  in  the  theory  of  the  latter  planet,  which  I  shall  men- 
tion hereafter,  and  have  not  yet  had  time  to  correct.  Practi- 
cally, therefore,  the  mass  of  Venus  is  determined  only  from 
observations  of  the  Sun  and  of  Mercury,  and  that  of  Mercury 
from  observations  of  Venus. 

Correction  of  equinox  and  equator. 

24.  t Could  all  the  observations  be  directly  referred  to  a 
visible  equinox  and  equator,  the  corrections  above  enumerated 
would  have  been  the  only  ones  which  it  was  necessary  to 
include  in  the  equations  of  condition.  But,  as  a  matter  of 
fact,  the  observations  were  all  referred  to  an  assumed  system 
of  Right  Ascensions  and  Decimations  of  standard  stars — my 
own  system  in  Eight  Ascension  and  Boss's  in  Declination. 
We  must  therefore  introduce  two  additional  unknowns  into 
the  equations,  which  I  have  represented  in  the  following  way: 

<*,  the  common  error  of  the  adopted  Right  Ascensions. 
#,  the  common  error  of  Boss's  Declinations. 

The  first  quantity  will  appear  only  in  the  equations  derived 
'from  observed  Right  Ascensions  and  the  second  only  in  the 
equations  derived  from  Declinations,  the  coefficient  being  unity 
in  each  case. 


24]  CORRECTION  OF  EQUINOX  AND  EQUATOR.  47 

That  the  value  of  6  found  in  this  way  should  be  regarded 
as  a  correction  to  the  Declinations  of  the  equatorial  stars  will 
appear  by  the  following  considerations.  The  mean  heliocen- 
tric orbit  of  a  planet  as  projected  on  the  celestial  sphere  is 
undoubtedly  a  great  circle.  On  the  other  hand,  in  view  of  the 
systematic  discordance  always  found  to  exist  in  measures  of 
absolute  Declinations  near  the  equator,  and  of  the  fact  that 
these  absolute  Declinations  depend  upon  assumed  constants 
and  laws  of  refraction,  which  are  necessarily  affected  with 
greater  or  less  uncertainty,  and  are  otherwise  subject  to 
systematic  errors,  instrumental  or  personal,  of  an  obscure 
character,  but  strongly  shown  by  a  comparison  ot.the  Declina- 
tions derived  from  the  work  of  different  observatories,  it  can 
not  be  assumed  that  these  Declinations  are  free  from  sys- 
tematic error.  JSow,  m  one  circle  ot  Decimation,  say  the 
equator,  we  may  expect  that  the  error  will  be  nearly  constant 
around  the  sphere,  since  the  causes  of  error  will  generally  be 
nearly  constant  for  any  one  Declination.  This  conclusion  is 
confirmed  by  a  comparison  of  the  best  star  catalogues. 
Moreover,  between  the  zodiacal  limits,  the  error  in  each  par- 
ticular case  is  not  likely  to  differ  very  greatly  from  the  error 
at  the  equator.  Even  if  the  difference  should  be  considerable 
the  various  values  of  the  error  of  the  different  Decimations 
must  have  a  certain  mean  value,  so  that  in  the  case  of  each 
particular  star,  or  each  region  of  the  heavens,  we  may  conceive 
the  actual  error  to  be  divided  into  two  parts — one  the  mean 
value  in  question,  and  the  other  the  deviation  from  this  mean. 
The  latter  is  probably  smaller  than  the  former,  and  in  any 
case  can  not  very  well  be  determined  from  observations  of  the 
planets.  But  the  condition  that  the  planet  moves  on  a  great 
circle  of  the  sphere  admits  of  the  mean  value  being  deter- 
mined with  great  precision.  It  should,  therefore,  be  included 
in  the  equations  of  condition. 

The  value  of  <*,  the  common  error  of  all  the  Eight  Ascen- 
sions, can  obviously  not  be  determined  from  the  equations  in 
.Eight  Ascension  alone,  because  the  only  result  that  such 
observations  can  give  us  would  be  the  values  of  the  Eight 
Ascensions  referred  to  some  assumed  equinox.  The  coefficient 
of  a  would  therefore  completely  disappear  from  the  equations 


48   '  MERCURY,   VENUS,   AND  MARS.  [24 

of  condition  in  Eight  Ascension.  But  since  the  same  unknown 
quantities  are  introduced  into  the  equations  of  condition  in 
Eight  Ascension  and  in  Declination,  the  requirement  that  the 
two  sets  of  equations  shall  give  common  values  of  these 
quantities  does  away  with  this  indetermiriatiou  and  enables 
determinate  values  to  be  found.  In  fact,  this  method  does  not 
differ  in  principle  from  that  usually  adopted,  in  deriving  the 
Eight  Ascensions  of  stars  from  observations  of  the  Sun.  The 
latter  consists  in  deriving  the  Sun's  absolute  longitude  from 
observations  of  its  Declination  and  absolute  Eight  Ascensions 
of  the  stars  by  comparing  them  with  the  Sun.  In  the  same 
way  we  may. consider  that,  in  observations  of  the  planet,  the 
Sun's  absolute  longitude  is  derived  from  observations  of  Decli- 
nations of  the  planet,  and  then  a  comes  out  from  the  observa- 
tions in  Eight  Ascension. 

I  have  deemed  it  absolutely  necessary  that  all  the  equations 
of  condition  should  be  solved  by  the  method  of  least  squares. 
By  this  method  alone  can  the  results  of  the  observations  as 
regards  separate  values  of  the  elements  and  constants  be  prop- 
erly brought  out.  But  the  work  of  constructing  and  solving 
a  system  of  nine  thousand  equations  of  condition,  each  involv- 
ing twenty  unknown  quantities,  would  be  extremely  laborious, 
and  might  even  require  a  century  for  its  completion,  if  done  in 
the  usual  way.  It  was  therefore  necessary  to  adopt  every 
device  by  which  the  labor  could  be  reduced  to  a  minimum. 
One  device  was  the  dropping  of  all  superfluous  decimals  in  the 
coefficients  of  the  equations.  Since  tbe  errors  thus  produced 
would  be  purely  accidental,  it  follows  that  if  the  sum  of  the 
products  obtained  by  multiplying  the  value  of  each  unknown 
quantity  by  the  error  of  its  coefficient  in  the  equation  of  con- 
dition is  but  a  small  fraction  of  the  necessary  probable  error 
of  the  absolute  term,  no  serious  harm  will  result  from  the 
errors  of  the  coefficients. 

Another  device  was  the  construction  of  tables  for  finding 
the  coefficients.  Such  tables  relating  to  Mercury  and  Venus 
are  found  in  Vol.  II,  Part  1,  of  the  Astronomical  Papers. 
These  tables  are,  however,  only  given  for  one  mean  anomaly  in 
each  case,  and  therefore  require  computations  dependent  on 
the  value  of  the  other  anomaly.  They  were  therefore  extended 


24,  25]        INTRODUCTION  OF  SECULAR  VARIATIONS.  49 

to  tables  of  double  entry,  so  that  the  value  of  the  derivatives 
of  the  geocentric  Eight  Ascension  or  Declination  at  any  epoch 
could  be  taken  from  the  tables  at  sight.  The  arguments  were 
the  mean  anomaly  of  the  planet  and  the  day  of  the  year  at 
which  the  planet  last  passed  through  its  perihelion. 

Introduction  of  the  secular  variations. 

25.  When  the  equations  of  condition  are  formed  on  the  plan 
just  set  forth,  the  unknown  quantities  will  be  the  corrections 
to  the  elements  or  to  the  mean  longitude  at  the  date  of  each 
equation.  But  every  one  of  the  unknown  quantities  which 
have  been  enumerated,  the  correction  of  the  masses  excepted^ 
is  subject  to  a  secular  variation.  Hence,  instead  of  the 
unknown  quantities  heretofore  denned,  we  introduce  two 
others,  the  one  the  value  of  this  unknown  at  some  assumed 
mean  epoch,  which,  for  reasons  already  set  forth,  must  first 
be  determined  from  the  observations;  the  other  the  secular 
variation  in  a  unit  of  time.  The  unknown  quantities  which 
have  been  enumerated  make  twelve  for  each  equation  of  con- 
dition. Eleven  of  these  are  subject  to  a  secular  variation,  so 
that  if  the  secular  variations  were  introduced  into  the  original 
equations  of  condition  they  would  each  have  twenty-three 
unknown  quantities. 

The  following  device  was  employed  to  reduce  to  a  minimum 
the  work  of  introducing  and  determining  the  secular  variations 
of  the  various  elements : 

Firstly,  the  whole  time  covered  by  the  observations  was 
divided  into  periods,  never  exceeding  ten  years,  except  when 
the  observations  were  very  few  in  number,  or  entitled  to  but 
small  weight.  It  was  then  assumed  that  no  error  would  arise 
from  supposing  the  value  of  the  unknown  quantity  to  be  the 
same  throughout  the  period  as  it  was  at  the  mid-epoch  of  the 
period.  The  maximum  absolute  error  thus  arising  would  be 
the  secular  variation  during  half  the  length  of  the  period,  and 
the  mean  error  the  secular  variation  during  one-fourth  of  the 
period;  but  actually  the  effect  of  even  this  error  would  be 
almost  entirely  nullified  by  the  combination  of  positive  and 
negative  coefficients  throughout  each  period. 
5690  N  ALM 4 


50  MERCURY,   VENUS,   AND  MARS.  [25 

Let  us  now  put 

x,y,    •    •    • 

the  corrections  to  the  elements  at  any  epoch,  T. 

Let 

a  x+  b  y  +  cz  -f  .    .     .  ^=n 

be  an  equation  of  condition  between  these  quantities  at  this 
epoch.  From  a  system  of  such  equations,  extending  through  a 
period  numbered  i,  during  which  #,  #,  etc.,  may  be  considered 
as  constant,  we  derive  normal  equations  of  the  form  — 

[aa]tx+[db]ty  +     .    .    .     =  [an], 


which  I  shall  call  partial  normal  equations,  and  which  we 
might  solve  so  as  to  obtain  the  values  of  x,  y,  etc.  This  solu- 
tion is  not,  however,  necessary.  The  values  of  the  unknown 
quantities  being  really  of  the  general  form  — 


we  may  imagine  these  values  substituted  in  the  normal  equa- 
tions (1),  the  value  T,  of  t  for  the  mean  epoch  of  the  period 
being  substituted  for  t. 

Let  us  now  suppose  that  we  introduce  the  quantities  #0,  2/o,  •  •  > 
#',  y',  .  .  into  the  original  equations  of  condition,  using  for  t 
the  value  rtj  which  pertains  to  the  mean  epoch  of  the  period. 
Our  equation  of  condition  will  thus  become  — 


axQ  +  fy/o  4-     •     •     +  artxf  +  brty'  +     .     .     =  n        (3) 

If  from  a  system  of  conditional  equations  of  this  form  we 
form  the  normal  equations  for  all  the  unknown  quantities,  the 
results  will  be  these  : 

Partial  normal  equation  in  x0  ; 

[aa]f#0-f-  [db]ty0  +  .  .  +  rJaaJX  +  r^ab^y1  +  .  .  =  [aw],  (4) 


25]  INTRODUCTION  OF  SECULAR  VARIATIONS.  51 

Partial  normal  equation  in  x1  j 

T,[aa],a?o  +  T,[a&],y0  +  .    .  +  T«2[aa],#'  +  rf  [ab^y1 

+  .  .  =  rt  [an]t       (5) 

We  conclude  that  the  partial  normal  equations,  when  the  full 
number  of  unknown  quantities  is  included,  may  be  derived 
from  those  of  the  form  (1)  by  the  following  rules. 

(1)  Each  partial  normal  equation  in  XQ,  y0,    .    .     .  is  formed 
from  that  in  #,  y,  etc.,  by  adjoining  to  the  first  member  of  the 
equation  the  member  itself  multiplied  by  r  and  then  changing 
x,  y,  .     .    .to  x0j  XQ'J  and,  in  the  products  by  r,  changing 
x,  y,     .     .     .    into  a?',  y',     .     .     . 

(2)  The  partial  normal  equation  in  a?',  y',  .     .     .   is  formed 
from  the  partial  equation  in  x0j  y^  .     .    .  by  multiplying  all 
the  terms  throughout  by  the  factor  r. 

The  final  or  complete  normal  equations  in  all  the  unknown 
quantities  being  formed  by  the  addition  of  the  partial  normals, 
the  formulae  for  the  coefficients  are  as  follow  : 

For  the  final  equation  in  XQ 

[aa]  =      [aa]!  +       [aa],  +     .     .     .     +       [aa]n 

[ab]  =      [a&]i+       [a&]2+     .     .     .     +       [ab]n 

[aa]'  =  n  [aa]i  +  r2  [aa]2  +     .    .    .     +  rn  [aa], 
[an]    =       [an]^       [an]2+     .    .    .    +       [aw], 

For  the  final  equation  in  x' 

[aa]"  =  r,a  [aa]!  +  r,2  [oa]2  +     .     .     .     +Tn»[aa]n 

.     .     .     +rn*[ab]n 


[an]"  =  n  [an]i  +  T2  [an]2  +     .     .    .     +  rn  [an]n 

The  final  equations  for  all  the  unknown  quantities  will  then 
be  of  the  form 

[oa]  x0  +  [aft]  y0  +     .     .     +  [aa]1  x'  +  .     .     .  =  [an] 

...     (8) 
[aa]'xo+[ab]/y()+  .    .    .   +[aa]"0/+  .    .     .  =  [an]" 


52  MERCURY,   VENUS,   AND  MARS.  [25,26 

The  epoch  from  which  we  count  the  time,  r,  is  arbitrary. 
An  obvious  advantage  will  be  gained  in  counting  it  from  the 
mid- epoch  of  all  the  observations.  Then  we  shall  have,  by 
putting  w^  w2,  etc.,  for  the  sum  of  the  weights  for  the  different 
periods  : 

MI  r\  +  w2  r2  +     •     •     •     +  wn  rn  =  0  (9) 

If  the  observations  are  then  equally  distributed  around  the 
orbits  of  the  planet  and  of  the  Earth  it  may  be  expected  that 
the  coefficients 

[««.]',  [aft]'    ....  (10) 

will  all  nearly  or  quite  vanish.  Practically  we  may  expect  that 
as  observations  are  continued  through  successive  revolutions 
the  ratios  of  these  to  the  other  coefficients  will  approach  zero 
as  a  limit.  We  may  then  divide  the  normal  equations  into  two 
sets,  one  containing  the  quantities  x0,  y^  etc.,  and  the  other 
#',  y',  etc.  The  coefficients  (10)  being  small,  the  two  sets  of 
normals  will  be  nearly  independent,  and  we  may  omit  the 
terms  (10)  in  the  first  approximation,  and  introduce  them  in 
one  or  two  successive  approximations  so  far  as  necessary. 

The  unit  of  time  is  also  arbitrary.  A  certain  advantage  in 
symmetry  will  be  gained  by  so  choosing  it  that  the  mean  value 
of  T3  shall  not  differ  greatly  from  unity.  It  was  found  that 
twenty-five  years  was  a  sufficiently  near  approximation  to  be 
adopted  for  all  three  planets. 

Dates  and  weights  for  epochs  and  periods. 

26.  As  want  of  space  makes  impracticable  the  present  publi- 
cation of  the  great  mass  of  material  worked  up,  the  following 
particulars  have  been  selected  as  those  most  likely  to  be  use- 
ful in  judging  and  criticising  the  work.  We  give  three  tables, 
showing  the  division  of  the  dates  of  observation  into  periods, 
and  the  weights  for  each  period.  The  first  column  of  each 
table  contains  the  number  or  designation  of  the  period,  as 
found  in  the  manuscript  books.  The  second  contains  the 
mean  year  of  the  period.  The  third  column  shows  the  time 


26]         DATES  AND  WEIGHTS  FOR  EPOCHS  AND  PERIODS.          53 

of  this  mean  period  from  the  mid-epoch  of  the  observations, 
which  is  taken  as  follows  : 

For  Mercury,  1865.0 
Venus, 


The  next  column  contains  the  sum  of  the  weights  of  the 
equations  in  each  period,  as  used  'in  forming  the  normal  equa- 
tions. These  were  not,  however,  the  weights  actually  used 
in  multiplying  the  coefficients  of  the  equations  of  condition. 
Owing  to  the  diversity  in  the  quality  of  the  observations  at 
different  times  it  was  not  found  convenient  to  reduce  the 
equations  at  once  to  a  uniform  system  of  weights,  and  so  dif- 
ferent units  of  weight  were  selected  for  the  older  observations 
and  for  the  earlier  observations.  After  the  partial  normal 
equations  were  formed  they  were  multiplied  by  the  factor  F, 
necessary  to  reduce  them  to  a  standard  in  which  the  unit  of 
weight  should  correspond  to  the  mean  error  — 


The  sums  of  the  weights  reduced  by  these  factors  are  shown 
in  the  table. 

In  arranging  the  weights  and  selecting  the  factors  it  should 
be  remarked  that  a  liberal  allowance  was  made  at  each  step 
for  probable  constant  errors,  which  results  in  the  given 
weights  being  much  smaller  than  they  would  have  been  by 
the  theoretical  treatment  of  the  original  observations.  Not- 
withstanding this  allowance  the  final  result  seems  to  show 
that  it  was  still  insufficient,  and  that  the  actual  weights  of 
the  results  are  less  than  would  follow  even  from  the  final  ones 
as  given.  . 

The  partial  normal  equations  for  each  period  after  being' 
multiplied  by  the  factors  F,  are  added  to  form  the  final  normal 
equations  as  derived  from  meridian  observations. 


54  MERCURY,  VENUS,  AND   MARS.  [26 

WeightSj  epochs,  and  periods  of  partial  normal  equations. 

MERCURY. 


1 

1 

Right  Ascension. 

Declination. 

Mean 
year. 

T                Wt 
(units  of  257.) 

F- 

Mean 

year. 

T 

(units  of  25  y. 

Wt. 

F. 

I 

2 

3 
3i 

3-2 

4 
5 
5i 

8- 

6, 
6, 

8 
9i 

9-2 
10, 

I02 

«i 

n2 
Ii3 

1766.60 

1784.  22 

1799.81 

-3.  9360 
-3.2312 

—2.  6076 

26!  I 

* 
1 

1765-50 

1782.99 

-3.  9800 
—3.  2804 

• 

O.  2 
4.9 

1 
f 

1796.42 
1802.37 
1809.  1  8 
1824.  83 

—2.  7432 
-2.5052 
—2.  2328 
—I.  6068 

5-0 

39-9 
52.8 

74-1 

I 

I 

1809.  53 

—  2.  2188 

18.9 

i 

1818.  79 
1825.  80 
1835-56 

—  1.8484 
—  1.5680 
—  I.  1776 

0.9 
34.5 

75-o 

i 
i 
i 

1833.84 
1838.  26 
1843.  97 

1855-92 
1862.  79 
1867.  18 
1872.64 
1877.05 
1882.  17 
1886.  29 
1889.  70 

I  .  2464 

75-3 
141.5 

281.5 
201.  5 

189.5 

294.5 

214.0 

204.5 
171.5 

338.  o 
176.0 

1% 
1 

1 
1 

1 
1 
1 

I   0606 

1843.  74 
1855.90 
1863.  10 
1867.  12 
1872.  62 
1877.12 
1882.  24 
1886.29 
1889.82 

—  o.  8504 
—  o.  3640 
—  o.  0760 
+o.  0848 
4-o.  3048 

-j-o.  4848 
4-o.  6896 

-j-o.  8516 
-j-o.  99*28 

98.8 

83-3 
99-8 
1  86.  o 
129.8 
129.8 
108.2 
199.8 
109.5 

i 

i 
i 

* 
i 

| 

—  0.8412 
—o.  3632 
—o.  0884 
-j-o.  0872 
-l-o.  3056 
-j-o.  4820 
-fo.  6868 
4-0.8516 
-j-o.  9880 

VENUS. 


I 

1755.83 

—4.  2868 

"•3 

* 

1759.69 

—4.  1324 

7.0 

i 

2 

1767.92 

—3.  8032 

19.7 

i 

1770.  18 

-3.7128 

IO.  O 

i 

3 

1781.06 

—3-  2776 

3-7 

i 

I793.25 

—  2.  7900 

13.5 

i 

4 

1792.47 

—2.8212 

12.3 

i 

1806.  73 

—2.  2508 

65.5 

* 

5 

1802.  64 

—2.4144 

23-3 

i 

i8i5-59 

—  1.8964 

67.5 

i 

6 

1810.  31 

—2.  1076 

34-0 

i 

1823.75 

—  1.5700 

197.0 

i 

7 

1816.  88 

—  1.8448 

42.  7 

i 

1836.  02 

—  1.0792 

762.  o 

8 

1825.55 

—  I.4980 

141.0 

1844.08 

—o.  7568 

650.  o 

9 

1835.31 

—  I.  1076 

339-3 

i 

1854.  24 

—o.  35°4 

333-o 

10 

1843.98 

—o.  7608 

259.3 

i 

1861.43 

—o.  0628 

749.0 

li 

1853-51 

—o.  3796 

205.3 

i 

1868.06 

-j-o.  2024 

815.0 

12 

1861.  60 

—  o.  0560 

353-7 

* 

1875-32 

-j-o.  4928 

692.0 

13 

1868.  12 

+o.  2048 

466.  o 

1 

1883.  15 

4-o.  8060 

819.  o 

i 

H 

1875-  38 

+o.  4952 

399-5 

1 

1888.56 

4-1.0224 

801.0 

i 

1C 

1883.  09 

-j-o.  8036 

04.  c 

i 

16 

1888.  67 

-j-I.  0268 

D   T^   J 

520.  5 

2 

| 

*/     *? 

26,27]  UNKNOWN  QUANTITIES  OF  EQUATIONS.  55 

Weights,  epochs,  and  periods  of  partial  normal  equations. 

MARS. 


Right  Ascension. 

Declination. 

T3 

o 

1 

Mean 
year. 

T 

(units  of  25  y.} 

Wt. 

F. 

Mean 
year. 

r 

(units  of  25  y.} 

Wt. 

F. 

, 

1757-43 

—3-  9428 

25-3 

i 

1758.82 

-3.8872 

8.8 

i 

2 

1770.55 

—3-  4i8o 

II.  0 

* 

11773-79 

-3-  2884 

8.8 

3 

1787.82 

-2.7272 

10.  0 

* 

1794.48 

—2.  4608 

13.0 

i 

4 

1799.77 

—  2.  2492 

20.7 

1 

1804.  91 

-.-2.  0436 

47.0 

i 

5 

1811.32 

-1.7872 

14.7 

* 

!  1813.00 

—  I.  7200 

30.5 

1 

6 

1829.  17 

—  1.0732 

60.  o 

i 

1828.04 

—  I.  Il84 

93-o 

7 

1837-  39 

—o.  7444 

121.  O 

i 

1837.  18 

—o.  7528 

371.0 

8 

1845-  39 

—o.  4244 

76.3 

t 

1844.95 

—o.  4420 

255-0 

9 

1853-36 

—  o.  1056 

90.  o 

* 

1853.  02 

—o.  1192 

245.0 

10 

1861.  07 

-j-o.  2028 

114.  o 

i 

1860.  94 

+0.  1976 

306.0 

ii 

1869.  20 

+o.  5280 

124.  o 

* 

1868.  80 

-f  o.  5120 

197.0 

12 

1877.71 

-j-o.  8684 

132.  o 

i 

1877.  38 

+o.  8552 

257.0 

J3 

1883.  27 

4-i.  0908 

91.  o 

i 

1883.26 

+1.0904 

1  60.  o 

14 

1888.  85 

+  1.3140 

115.5 

i 

1888.  48 

+  1.2992 

167.0' 

Unknown  quantities  of  the  equations. 

27.  For  convenience  in  solving  the  equations  of  condition 
the  coefficients  of  the  equations  were  multiplied  by  such 
numerical  factors  as  would  reduce  their  general  mean  abso- 
lute value  to  numbers  of  approximately  the  same  order  of 
magnitude.  Hence,  the  unknown  quantities  themselves  are 
not  the  corrections  to  the  elements,  but  these  corrections 
divided  by  the  adopted  factors. 

In  the  case  of  Mercury  the  absolute  term  was  also  multi- 
plied by  10,  so  that  effectively  the  factors  in  question  were 
reduced  to  one-tenth  part  of  their  value.  The  unknown 
quantities  of  the  equations  are  represented  by  the  symbols 
of  the  elements  to  which  they  relate  inclosed  in  brackets. 

For  convenience  of  reference  the  following  table  is  given, 
showing  the  factors  used  in  the  case  of  each  planet.  In  the 
case  of  Mercury  the  column  (a)  shows  the  factors  by  which  the 
differential  coefficients  were  actually  multiplied;  (b)  the  factor 
by  which  the  unknown  quantity,  as  finally  found,  must  be 


56  MERCURY,  VENUS,  AND  MARS.         [27,  28 

multiplied  to  obtain  the  correction  as  expressed  in  the  last 
column..  In  the  case  of  Venus  and  Mars  these  factors  are  the 
same. 

Factors  by  which  the  unknown  quantities  are  to  be  multiplied  to 
obtain  corrections  of  the  elements. 


Symbol  of 
unknown. 

Mercury 

Factor  ior  — 
Venus. 

.  Mars. 

Corr.  of 
element. 

w 

(*) 

[  »»] 

1 

0.1 

7 

0.3 

dm  :  mQ 

(  *  ] 

40 

4 

5 

2 

dl 

1  JJ 

30 

3 

6 

2.5 

dJ 

[Nj 

30 

3 

7 

2.5 

sin  JdR 

I  «  ] 

30 

3 

3 

10^-7 

de 

f  *  ] 

100 

10 

439 

100-r7 

d?r 

1  *  ] 

100 

2.056 

3 

1.3323 

edn 

1  *  1 

10 

1 

4 

4 

de 

[«"] 

6 

0.6 

2.5 

2 

de" 

[«"-] 

6 

0.6 

2 

2 

e"dn" 

I  «  1 

10 

1 

1 

5 

Of 

[  *  ] 

10 

1 

5 

5 

d 

[I"] 

10 

1 

4 

3 

dl" 

The  secular  variation  of  each  unknown  in  25  years  is 
expressed  sometimes  by  a  suffixed  1,  sometimes  by  an  accent, 
thus: 

[1]'  =  [l]i  =  change  of  [I]  in  25  years. 

28.  It  may  also  be  useful  to  give  the  values  of  the  principal 
coefficients  in  each  of  the  normal  equations.  They  are  found 
in  the  following  table.  Were  the  other  coefficients  all  zero, 
these  numbers  would  indicate  the  weights  of  the  different 
unknown  quantities  as  resulting  from  the  solution.  Several 
of  them  were  greatly  diminished  by  the  process  of  solution. 


28,  29] 


ORDER   OF  ELIMINATION. 


57 


Values    of   the  principal  diagonal  coefficients  in  the  normal 

equations. 


I 

rtercury. 

Venus. 

Mars. 

Symbol  c 

f 

From 

oefficien 

:. 

From  mer. 
observa- 

From 
transits. 

Sum. 

From  mer. 
observa- 

From 
transits. 

Sum. 

mer. 
observa- 

tions. 

tions. 

tions. 

mm 

5488 

o 

5488 

^868 

2929 

8797 

17887 

11 

I0559 

11308 

21867 

598i 

3540 

9521 

20924 

"  JJ 

15222 

1296 

16518 

13232 

7444 

20676 

28783 

:  NN  ; 

14176 

2304 

16480 

I795I 

1636 

19587 

32478 

-  ee  '. 

19015 

5076 

24091 

5686 

3350 

9036 

20119 

7T  7T 

8621 

8352 

16973 

5290 

1732 

7022 

20564 

£  £ 

IIOOI 

196 

11197 

11429 

3598 

15027 

31460 

''  e"  e"  '' 

9757 

508 

10265 

9586 

665 

10251 

15909 

'IT"  TT" 

9099 

261 

9360 

5836 

1895 

7731 

14911 

'•  r//  r// 

^242 

o 

5242 

'  /"/"  = 

Or 

13041 

542 

13583 

11031 

2349 

I338o 

15427 

aa 

13230 

0 

13230 

335 

o 

335 

25138 

r  66  '' 

24657 

0 

24657 

15196 

o 

15196 

53975 

11  ' 

7014 

67155 

74169 

6005 

8983 

14988 

26689 

JJ  " 

12366 

9383 

21749 

9837 

13014 

22851 

23440 

;  NN  ; 

"°35 

16682 

27717 

14724 

2874 

17598 

29494 

ee  ' 

15437 

29647 

45084 

5743 

8610 

'4353 

24364 

7T7T 

6745 

493  i  8 

56063 

4948 

4483 

943i 

27131 

ee 

8488 

1418 

9906 

8458 

6306 

14764 

25675 

:  e"  eff  " 

8409 

2937 

11346 

9805 

1682 

11487 

22947 

:TT"  TT//= 

8439 

1513 

9952 

5242 

4805 

10047 

17356 

V'r'/n 

5432 

o 

5432 

'.  l"  l"  ". 

11629 

3126 

I47S5 

10677 

5667 

16344 

20655 

aa 

11400 

o 

11400 

297 

o 

297 

33624 

;  66  \ 

18716 

o 

18716 

10772 

o 

10772 

42405 

NOTE. — The  coefficients  for  Mercury  and  Venus  in  this  table  are  given  as  they 
were  used  in  the  solution,  after  dropping  the  units  from  all  the  terms  of  the 
equations,  except  those  from  transits  of  Mercury. 

Order  of  elimination. 

29.  In  dealing  with  so  extensive  a  system  of  unknown 
quantities  it  is  impracticable  to  investigate  the  dependence  of 
each  upon  all  the  others.  It  is  therefore  essential  to  arrange 
the  unknowns  in  an  order  partly  that  of  interdependence  and 
partly  that  of  the  liability  of  each  to  subsequent  change  by 
discussion  and  adjustment.  Hence,  the  mass  of  the  planet. 
Mercury  or  Venus,  should  be  first  eliminated,  as  being  that 
unknown  which  is  least  affected  by  changes  in  the  final  values 
of  the  other  unknowns.  The  secular  variations,  as  derived 


58  MERCURY,  VENUS,  AND  MARS.         [29,  30 

from  meridian  observations,  are  nearly  independent  of  the 
corrections  to  the  other  elements.  The  solar  elements  are  to 
be  subsequently  determined  by  a  combination  of  the  results 
of  the  observations  of  the  Sun  and  of  the  three  planets. 

Guided  by  these  considerations,  the  order  of  elimination 
was,  with  some  exceptions,  as  follows  : 

1.  The  mass  of  the  disturbing  planet. 

2.  The  five  elements  of  the  observed  planet. 

3.  The  four  elements  of  the  Earth's  orbit. 

4.  The  corrections  to  the  star-positions  for  the  mid-epoch. 

5.  The  secular  variations  of  the  eleven  quantities  (2),  (3), 
and  (4),  taken  in  the  same  order. 

Treatment  of  meridian  observations  of  Mercury. 

30.  In  the  case  of  Mercury  the  factors  of  the  coefficients  of 
the  equations  were  chosen  large  enough  to  admit  of  the  deci- 
mals being  dropped  from  the  products  without  prejudice  to 
the  accuracy  of  the  final  result.  This  was  done  to  facilitate 
the  formation  of  the  normal  equations.  For  the  same  reason 
the  factors  were  made  so  small  that  the  absolute  numerical 
values  of  the  coefficients  should  generally  not  exceed  13.  As 
this  degree  of  precision  is  far  short  of  that  usually  employed 
for  correcting  the  elements  of  a  planet,  it  may  be  well  to  set 
forth  the  considerations  on  which  it  is  based. 

Let  any  equation  of  condition  as  actually  used  be  — 

ax-\-  by  +  cz  +     .     .     .     =n  (a) 

Let  the  coefficients  a,  &,  etc.,  be  affected  by  the  mean  errors 
e,  «',  etc.,  so  that  the  true  equation  should  be  — 


.     .     .     =  n 
This  true  equation  may  be  written  in  the  form  — 

ax  +  by  +     .     .     .     =  n—£x  —  e'y  —     .     .     .          (b) 

We  may  regard  (b)  as  a  rigorous  equation,  in  which  the  error 
of  the  second  member  is  increased  by  the  quantity  — 


30]  MERIDIAN   OBSERVATIONS   OF  MERCURY.  5£ 

and  the  only  effect  upon  tlie  precision  of  the  results  will  be 
that  arising  from  this  increased  probable  error.  Let  us  esti- 
mate its  magnitude.  From  an  examination  of  the  tables  used 
in  finding  the  coefficients  I  infer  that  the  probable  error  of  the 
coefficient  of  n  Avas  ±  1,  and  that  of  all  the  other  coefficients 
±  0.6.  The  mean  value  of  the  unknown  quantities  was  gener- 
ally a  small  fraction  of  a  second.  We  conclude,  therefore, 
that  the  probable  or  mean  value  of  the  error 

±  ex  ±  fy  i     .     .     . 

would  in  any  case  be  only  a  small  fraction  of  a  second.  More- 
over, these  errors  would  be  purely  accidental  and  not  system- 
atic, since  the  intervals  of  time  between  the  equations  were 
generally  so  long  that  the  coefficients  for  different  equations 
came  from  different  tables,  so  that  no  error  from  omitted  deci- 
mals in  any  one  equation  would  enter  into  the  other  equations. 

Now,  in  view  of  the  necessary  systematic  errors  which  affect 
observations  of  the  planets,  there  is  no  hope  of  approximating 
to  this  degree  of  accuracy  in  the  second  members  of  the  equa- 
tions. Were  the  observations  rigorously  correct  and  the 
values  of  the  unknown  quantities  finally  determined  affected 
by  no  error  except  that  arising  in  this  way,  they  would  be 
many  times  more  accurate  than  we  can  hope  to  make  them. 
The  errors  might,  in  fact,  be  considered  unimportant  in  the 
present  state  of  astronomy. 

It  has  already  been  remarked  that  the  scale  of  weights  was 
so  taken  that  the  unit  of  weight  should  correspond  approx- 
imately to  a  supposed  mean  error  i  1".0  in  the  value  of  each 
absolute  term  of  an  equation  of  condition,  so  far  as  the  error 
could  be  determined  from  the  discordance  of  the  original 
observations.  The  corresponding  probable  error  would  be 
dt  0".65.  In  the  case  of  Mercury,  however,  modifications  were 
made  which  prevents  this  mean  error  from  corresponding  to 
the  unit  of  weight  which  would  be  found  from  the  solutions  in 
the  usual  way.  In  the  first  place,  the  absolute  members  were 
all  multiplied  by  10;  in  other  words,  the  decimal  point  was 
dropped  from  tenths  of  seconds,  and  no  further  account  taken 
of  it.  Secondly,  in  consequence  of  the  probable  error  in  the 
coefficients  of  the  normal  equations  arising  from  the  irnperfec- 


•60  MERCURY,  VENUS,   AND  MARS.  [30 

tions  of  tlie  decimals,  the  final  values  of  these  coefficients 
would  be  subject  to  probable  errors  ranging  between  50  and 
100  units.  In  consequence  there  would  be  no  advantage  in 
retaining  the  last  figure  in  the  normal  equations,  and  it  was 
dropped  in  all  the  subsequent  solution  and  discussion  of  these 
equations. 

In  dropping  the  last  figure  from  the  absolute  term  of  the 
normal  equations  we  may  consider  that  we  are  merely  drop- 
ping the  tenths  of  seconds  and  that  the  units  are  once  more 
expressed  in  seconds.  Thus,  considering  only  the  effect  of 
this  operation,  the  unit  of  weight  would  correspond  to  a  mean 
error  of  ±  1.0  in  units  of  the  absolute  term.  But  in  dropping 
off  the  last  figure  from  the  coefficients  we  practically  reduce 
the  scale  of  weights,  considered  as  multipliers  of  the  equa- 
tions, to  one-tenth  of  their  former  value.  On  the  other  hand, 
in  expressing  the  unknown  quantities  in  terms  of  the  correc- 
tions to  the  elements,  we  divide  the  multipliers  by  ten,  so  that 
effectively  we  multiplied  the  coefficients  in  the  equations  of 
condition,  considering  the  unknown  quantities  to  be  defined 
as  on  page  56,  by  10.  Since  these  coefficients  are  of  the  second 
degree  in  the  normal  equations,  it  follows  that  the  scale  of 
weights  has  in  effect  been  increased  ten  fold.  Hence  the  unit 
of  weight  for  the  normal  equations  between  the  unknown 
quantities  as  finally  solved  will  correspond  to  the  mean  error 

€l  =  1.0  X    VI6  =  ±  3.1 

As  the  mean  error  is  at  best  a  rather  indefinite  quantity  in  a 
case  like  the  present,  we  may  consider  its  value  as  4  units  and 
even  then  as  by  no  means  rigorously  determined. 

Up  to  the  time  of  writing  no  attempt  has  been  made  to 
derive  rigorously  the  weights  of  the  unknown  quantities  from 
the  solution,  because  in  the  cases  of  most  of  the  uukowns  such 
weights  would  be  entirely  illusory.  The  fact  is  that  in  solving 
so  immense  a  mass  of  equations,  we  must  expect  systematic 
errors  to  vitiate  many  of  the  results.  The  observations  of 
Mercury,  especially  of  its  Eight  Ascension,  are  not  made  on 
.a  uniform  system  5  sometimes  the  limb  is  observed,  sometimes 
the  apparent  center  or  the  center  of  light. 


30,  31]  TRANSITS   OF  MERCURY.  61 

An  ideally  perfect  system  of  reduction  would  require  us  to 
reduce  each  separate  observation  with  a  semidiaineter  corre- 
sponding to  the  personal  equation  of  the  observer.  This  being 
entirely  impracticable,  we  must  regard  the  reduction  of  the 
observer's  semidiameter  to  that  used  in  the  reductions  as  a 
probable  error.  In  fact,  however,  it  will  be  of  a  systematic 
character,  varying  at  each  point  of  the  relative  orbit  of 
Mercury,  and  going  through  a  cycle  of  changes  impossible  to 
determine  in  a  synodic  period  of  the  planet.  It  is  impracti- 
cable to  give  even  a  full  discussion  of  these  errors;  we  shall, 
however,  meet  with  a  proof  of  their  magnitude. 

Introduction  of  the  equations  derived  from  observed  transits  of 

Mercury. 

31.  The  relations  between  the  elements  of  Mercury  and  the 
Earth  derived  from  this  source  are  shown  in  my  Discussion  of 
Transits  of  Mercury  (A.  P.,  Vol.  I,  Part  VI.)  On  page  447  are 
found  expressions  for  those  linear  functions  of  the  corrections 
to  the  elements  which  are  determined  by  the  November  and 
May  transits,  respectively.  With  a  slight  change  of  notation 
to  correspond  with  that  of  the  present  paper,  these  functions 
are  as  follows : 

V  =  1.487  61  -  0.487  dx  -  1.137  tie  -  1.01  dl"  -f  1.19  e"dn" 

+ 1.58  de" 
W  =  0.716  61  +  0.284  drt  +  0.896  de  -  0.97  61"  -  1.11  e"dn" 

-  1.62  de" 

The  values  of  V  and  W  being  derived  from  a  series  of  transits 
extending  from  1677  to  the  present  time,  enable  us  to  deter- 
mine both  these  quantities  at  some  epoch,  and  their  secular 
variations.  The  values  derived  from  the  transits,  together 
with  their  mean  errors,  are  found  on  page  460  of  the  work  in 
question.  Omitting  the  doubtful  factor  fc,  introduced  on 
account  of  a  possible  variability  of  the  Earth's  axial  rotation, 
which  was  not  proved  by  the  transits,  the  values  of  V  and  W 
were  found  to  be  as  follows : 

V  =  —  0".90  i  0".31  +  ( -  2//.63  ±  0';.59)  (T  -  1820) 
W  ==  +  0".84  ±  0".25  -f  (+  1".84  i  0".60)  (T  -  1820) 


62  MERCURY,   VENUS,   AND   MARS.  [31 

The  mean  epoch  for  the  transits  is  taken  as  1820,  to  which 
the  zero  values  correspond.  The  values  for  1865.0,  the  mid 
epoch  for  the  meridian  observations,  are,  therefore,  from  the 
transits  alone — 

V  =  -  2".08  ±  0".41 
W  =  +  1".67  ±  0".37 

This,  however,  is  only  a  first  approximation  to  the  quantities 
which  should  be  introduced.  Since  the  meridian  observations 
help  to  determine  the  values  of  V  and  W,  we  should  not 
regard  the  reductions  to  1865.0  as  final,  but  retain  the  results 
in  the  form  (a). 

Another  element  which  is  determined  from  the  observed 
transits  of  Mercury  with  greater  precision  than  it  can  be  from 
meridian  observations  is  the  longitude  of  the  node  of  the  orbit 
relatively  to  the  Sun.  In  the  paper  quoted  we  have  put — 

F  =  (30  -61"}  sin  i 
and  found  from  all  the  transits  up  to  1881, 

N  =  -  0".16  i  0".27  +  (0".28  dL  0".62)  (T  -  1820)        (b) 

The  values  of  Y,  W,  and  N,  found  from  the  discussion  in 
question,  give  rise  to  six  conditional  equations,  which  become 
completely  independent  when  we  take  as  observed  values  the 
secular  motions  and  the  absolute  values  at  the  mid-epoch  of 
observation.  This  mid-epoch  is  not  the  same  for  the  May  and 
November  transits.  But  I  have  assumed  that  no  serious  error 
would  be  introduced  by  taking  1820.0  as  the  epoch  for  all  three 
of  the  quantities,  Y,  W,  and  X. 

If  we  substitute  for  sin  i  66  its  value  in  terms  of  tfJ,  etc., 
namely, 

Sin  idB=-  0.6018  £J  +  0.796  sin  JtfN  +  0.721  de        (c) 

and  then  for  tf  J,  tfN,  tff,  their  values  in  terms  of  the  unknowns 
of  the  equations  of  condition,  we  shall  have 

N  =  -  1.805  [J]  +  2.394  [N]  +  0.721  [*]  -  0.122  [V]       (d) 


31]  TRANSITS  OF  MERCURY.  63 

Similar  expressions  will  be  found  for  the  values  of  Y  and  W 
by  substituting  for  the  corrections  to  the  elements  the  unknown 
quantities  of  the  conditional  equations,  as  already  given. 

Taking  1820.0  as  the  mid  epoch,  we  may  regard  the  inde- 
pendent quantities  given  by  the  transits  of  Mercury  to  be  the 
six  following  ones  : 


Vo  -  1.8  Yx  ;  W0  -  1.8  W,;  N0  - 

V,  ;          W,  5          N,     - 

Here  Y0,  W0,  and  N0  indicate  values  for  1865,  the  mid-epoch  of 
the  meridian  observations;  and  Y1?  W1?  and  Nj.  the  variations 
in  25  years.  The  six  conditional  equations  thus  found  from 
the  transits  may  be  written 

Yo  -  1.8  Yt  =  -  0".90  ±  0".31 
W0  -  1.8  Wt  =  +  0".84  i  0".25 

:NO  -  1.8  N!  =  -  0".16  ±  0".27 

Y,  =  -  0".66  i  0".15 

Wi  =  +  0".46  i  0".15 

Ni  =  +  0/7.07  ±  0/7.15 

Substituting  for  Y0,  YI,  etc.,  their  expressions  as  linear  func- 
tions of  the  unknowns  of  the  conditional  equations,  we  find 
the  following  six  equations,  which  are  to  be  used  as  conditional 
equations  additional  to  those  given  by  the  meridian  observa- 
tions : 

5.95  [1]  -  4.87  [TT]  -  3.41  [e]  -  1.01  [l"\  +  0.71  [n"\  +  0.95  [e"] 
-1.8)6.95[Z]!  -  4.87  [ir]i  —  3.41  [e],-  1.01  [l"]i+  0.71  [n"^ 
+  0.95[e//]1J  =  -O^O 

Weight  =  250 

2.86  [1]  +  2.84  [it]  +  2.69  [e]  -  0.97  [I11]  -  0.67  [n"\  -  0.97  [e"} 
-1.8  {2.86  [I],  +  2.84  [w-J!  +  2.69  (e},  -  0.97  [l"^  -  0.67  [n"}, 
-C.97^7'],}  =  +  07/.84 

Weight  =  300 

-  1.8  [J]  +  2.4  [N]  +  0.7  [f]  -  0.12  [I"} 
-  1.8  {  -  1.8  [  J]x  +  2.4  [NJi  +  0.7  [f  ]x  -  0.12  [///]l  }  =  -  0".16 

Weight  =  400 


64  MERCURY,  VENUS,   AND  MARS.  [31 


5.95  [1],  -  4.87  [TT]!  -  3.41  [e],  -  1.01  [l"^  +  0.71  [*"]i+  0.95 
=  -  0".66 

Weight  =  700 


2.86  [l]t  +  2.84  [TT]!  +  2.69  [e]i  -  0.97  [Z"]i  -  0.67  [TT"],  -  0.97  [a'7]! 
=  +  0".46 

Weight  =  700 

-1.8  [J],  +  2.4  [N]{  +  0.7  [e],  -  0.12  [I"},  =  +  0".07 
Weight  =  1,600 

The  weights  assigned  to  these  several  equations  have  been 
determined  by  the  following  considerations: 

We  have  already  found  that  in  the  equations  of  condition 
from  the  meridian  observations  as  finally  reduced,  the  scale  of 
weights  has  so  come  out  as  to  show  a  practical  mean  error  for 
weight  unity  of  about  ±  4".  Were  this  error  purely  accidental, 
the  weights  of  the  conditional  equations  derived  from  the 
transits  would  be  determined  in  the  same  way,  from  the  mean 
errors  assigned  to  them.  But,  as  a  matter  of  fact,  the  exist- 
ence of  systematic  errors  in  the  meridian  observations  is 
shown,  as  will  be  subsequently  explained,  by  the  large  value 
found  for  the  fictitious  quantity  6r2.  Since  observations  of 
transits  are  made  at  the  point  of  the  relative  orbits  of  Mercury 
and  the  Earth,  near  which  meridian  observations  are  rarely 
available,  and  are  of  a  higher  order  of  accuracy  than  meridian 
observations,  it  follows  from  the  theory  of  probabilities  that 
we  should  assign  a  larger  relative  weight  to  the  observations 
of  the  transits.  How  much  larger  does  not  admit  of  being 
determined  with  numerical  precision.  Actually  I  have  taken 
the  weights  as  if  the  mean  error  corresponding  to  weight 
unity  were  between  5  and  6.  In  the  case  of  the  motion  of  the 
node  a  still  larger  weight  has  been  assigned  to  the  secular 
variation,  from  the  belief  that  the  accuracy  of  the  determina- 
tion from  transits  relative  to  meridian  observations  is  in  this 
case  of  a  yet  higher  order  of  magnitude  than  in  the  case  of 


31,  32]        SOLUTION  OF  EQUATIONS  FOR  MERCURY.  65 

the  other  elements.    Whether  this  belief  is  justified  or  not 
must  be  left  to  the  decision  of  the  future  astronomer. 

The  first  three  of  the  preceding  six  conditional  equations 
may  be  treated  in  a  way  similar  to  that  adopted  for  the 
meridian  observations.  They  express  what  is  supposed  to  be 
equivalent  to  observations  of  the  three  quantities  V,  W,  and 
N  in  1820,  when  r  —  —  1.8.  Hence,  from  the  partial  normals 
in  the  six  principal  unknowns,  [e],  [>]...  [>"],  the  com- 
plete normals  may  be  formed  by  multiplication  by  r  and  i* 
(r  =  —  1.8)  in  the  way  set  forth  in  §  25. 

Solutions  of  the  equations  for  Mercury. 

32.  In  the  case  of  Mercury  and  Venus,  it  is  desirable  to 
know  to  what  extent  the  results  of  the  transits  diverge  from 
those  of  the  meridian  observations.  Hence,  as  already 
remarked,  two  solutions  of  the  equations  were  made,  termed 
A  and  B. 

Solution  A  is  that  derived  from  the  meridian  observations 
alone.  Solution  B  is  that  of  the  normal  equations  formed 
from  both  the  meridian  observations  and  the  transits. 

The  results  of  the  solutions  in  the  case  of  Mercury  are  shown 
in  the  following  tables.  The  relation  of  the  unknown  quan- 
tities given  in  the  first  columns,  A  and  B,  to  the  corrections 
of  the  elements  has  been  shown  in  a  preceding  section  (§  27). 
The  upper  half  of  the  table  shows  the  corrections  to  the 
elements;  the  lower  half  those  of  the  secular  variations. 

It  will  be  seen  that  all  the  values,  with  a  single  exception, 
come  out  less  than  a  unit.  In  stating  the  corrections  to  the 
elements,  it  must  be  remembered  that,  owing  to  the  proximity 
of  Mercury  to  the  Sun,  the  errors  of  geocentric  place  are  much 
less  than  those  of  the  heliocentric  elements,  so  that  an  error 
in  the  latter  indicates  a  proportionally  smaller  error  in  the 
actual  observations.  For  the  same  reason  we  must  expect  a 
less  degree  of  precision  in  the  elements  as  finally  derived  than 
in  the  case  of  the  other  planets. 
5690  N  ALM 5 


66  MERCURY,  VENUS,  AND  MARS 

MERCURY. 

Results  of  solutions  of  the  normal  equations. 


[32,  33 


Unknowns. 

EM 

Corrections  of  elements. 

Symbol. 

A. 

B. 

Symbol. 

A. 

B. 

[»"] 

—o.  1478 

—  o.  1207 

O.  I 

6  m  :  m 

—  o.  0148 

—  O.  OI2I 

'  / 

—o.  1342 

—  0.0752 

4- 

ii 

—o.  537 

—0.301 

:  j 

—o.  2436 

—  o.  2299 

3- 

6] 

—o.  731 

—  o.  690 

;N 

—o.  0227 

—0.  0201 

3- 

SinJdN 

-o.  068 

—o.  061 

t 

£ 

-fo.  2074 

+o.  2194 

i. 

6£ 

4-o.  207 

4-0.219 

e 

—  O.  I2O2 

4-0.  4094 

3- 

6e 

—  o.  361 

41.228 

'  TT 

4-0.  5209 

4-0.  2688 

10. 

6  7T 

+5-  209 

4-2.  689 

'  e" 

-f  o.  0669 

4-0.  8397 

0.6 

(5  ef/ 

-f-o.  040 

-f  o.  504 

V' 

—o.  2248 

—o.  7027 

0.6 

e"  6  TT// 

—  o.  135 

—  o.  422 

>// 

4-1.  1240 

4-1.0566 

2. 

6r» 

4-2.  248 

4-2.  113 

6 

—o.  2310 

—o.  2556 

I. 

6 

—0.231 

—  o.  256 

/" 

-0-0354 

—o.  0897 

I. 

61" 

-o.  035 

—  o.  090 

a 

4-0.  4803 

4-0.  4930 

I. 

a 

-fo.  480 

+o-  493 

/ 
J 

—  o.  2060 
—  o.  0114 

—  o.  1209 
-f-o.  0636 

1  6. 

12. 

Dt# 

—3.  296 

—  o.  137 

—i.  935 

-f  o.  764 

N; 

4~o.  looo 

4~o.  0930 

12. 

SInjDt<m 

-|-i.  200 

4-1.  116 

"  e  ~ 

4-o.  0681 

4-o.  0966 

4- 

Dt  6  e 

4-0.  272 

4-0.  386 

e 

—o.  1165 

+o.  0987 

12. 

Dt6e 

-i.  398 

4-1.184 

7T 

—o.  2385 

—  o.  0252 

40. 

Dtd7T 

—9-  540 

—1.008 

e" 

—  o.  1968 

+°-  I3I7 

2.4 

Dt  6  e// 

—o.  472 

4-0.316 

7T//" 

—o.  1677 

—o.  1193 

2.4 

e/s  Dt  6  TT// 

—  o.  402 

—o.  286 

r" 

4-o.  1108 

4-o.  0806 

8. 

Dtdr" 

4-0.  886 

4-0.  645 

6 

—o.  1826 

—o.  1233 

4- 

Dt<5 

—o.  730 

—o.  493 

I" 

—  o.  1442 

—0.3152 

4- 

Dtd/x/ 

-o.  577 

—  i.  261 

a 

—o.  3160 

—o.  1973 

4- 

Dta 

—  i.  264 

-o.  789 

Mean  epoch  of  corrections,  1865.0. 

Discordance  in  the  observed  Right  Ascensions  of  Mercury. 

33.  The  most  remarkable  feature  in  the  result  is  the  value 
of  the  quantity  represented  by  [r"\.  The  unknown  quantity 
introduced  with  this  symbol  had  as  its  coefficient  the  derivative 
of  the  geocentric  place  as  to  the  Earth's  radius  vector,  and  the 
result  would  therefore  be  an  apparent  constant  correction  to 
that  radius  vector.  Since,  however,  the  position  of  the  planet 
depends  only  on  the  ratio  of  the  distances  of  the  Earth  and 
Mercury,  it  follows  that  the  actual  correction  may  be  regarded 
as  a  correction  to  the  ratio  of  the  mean  distances. 

The  determination  of  the  mean  distances  by  KEPLER'S 
third  law  may  be  regarded  as  so  unquestionable  that  the  true 


33]  DISCORDANCE   OF   OBSERVATIONS.  67 

value  of  this  unknown  quantity  should  be  regarded  as  zero, 
and  the  result  as  a  purely  fictitious  one,  arising  from  errone- 
ous elements  of  reduction  or  systematic  personal  errors.  It 
was  the  possibility  of  the  latter  that  led  to  its  introduction. 
When  the  planet  is  east  of  the  Sun,  observations  are  always 
made  on  or  near  its  west  limb,  or  at  least  on  some  point  west 
of  the  true  center,  and  vice  versa.  The  value  of  dr"  therefore 
indicates  that  there  is  a  remarkable  systematic  difference  in 
the  observed  Eight  Ascension  according  as  the  planet  is  east 
or  west  of  the  Sun,  and  therefore  according  to  the  illuminated 
side.  The  sign  of  the  result  shows  that  the  reduction  to  the 
center  of  the  planet  was  apparently  too  small.  It  is  there- 
fore of  interest  to  learn  according  to  what  law  this  error 
changed  as  the  planet  moved  around  its  relative  orbit. 

It  has  up  to  the  present  time  been  impracticable  to  substi- 
tute the  unknown  quantities  in  the  original  equations  of  con- 
dition, and  thus  determine  the  separate  residuals,  and  for  the 
purpose  of  investigating  the  present  case  such  a  substitution 
is  the  less  necessary,  owing  to  the  sniallness  of  the  unknown 
quantities.  I  have  therefore  simply  determined  the  mean 
correction  to  the  Right  Ascension  given  by  all  the  observa- 
tions during  the  various  periods  in  six  segments  of  the  relative 
orbit,  near  the  elongations,  and  before  and  after  the  two 
conjunctions.  The  results  are  shown  in  the  following  table. 
Commencing  with  the  moment  of  inferior  conjunction,  column 
A  contains  the  mean  correction  to  the  tabular  Eight  Ascension, 
from  observations  made  within  about  twenty  days  following. 
Column  B  contains  the  observations  made  from  twenty  days 
after  the  inferior  conjunction  until  twenty  days  before  superior 
conjunction,  a  period  during  which  the  planet  was  generally 
near  its  greatest  west  elongation.  Column  C  contains  the 
observations  made  during  the  twenty  days  following  and  up 
to  superior  conjunction.  Then  follow  in  regular  order  the 
corresponding  results  when  the  planet  was  east  of  the  Sun, 
beginning  with  the  twenty  days  following  superior  conjunc- 
tion and  going  around  to  inferior  conjunction. 


68 


MERCURY,  VENUS,-  AND  MARS. 


[33 


Table  showing  the  mean  corrections  to  the  tabular  Right  Ascen- 
sion of  Mercury  in  six  segments  of  its  relative  orbit. 


Epochs. 

A 

B 

C 

1765-1791. 

"    wt. 

4-3.  24   4 

"    wt. 

4-2.  61   5 

"    wt. 

1793-1815.  

-(-2.06   6 

4-1.82  10 

-f  o.  97   4 

1817-1839  

+3-  06   6 

4-1.79  24 

-|-i.  13  24 

1840-1849 

+  1.46   6 

+  1.48  18 

—  o.  38  20 

1850-1859  
1860-1869. 

4-3-72   4 
4-1.  18  28 

4-0.  77  20 
+  1.14  72 

4-o.  08  1  6 
4-O.  31  44 

1870-1880  

+  i.  18  25 

4-o.  74  65 

—  o.  20  6  1 

1881-1892  

4-1.  19  38 

4-0.  98  63 

—o.  15  62 

D 

E 

F 

1765-1791  

"    wt. 
4-0.92  .  i.  5 

"    wt. 
•-{-1.30  10 

"    wt. 

+o.  81   3 

1793-1815.  

+2.  82   5 

4-i.  10  16 

+  1.85   5 

1817-1830 

4-O.  27   25 

4-3.  76  24 

—  i.  20   «; 

1  84.0—  1  840 

j  O  22   22 

—  o  S7  30 

j_o  71;   3 

i8co—  18^0 

J-O  60   14. 

—  O  7Q   28 

—  O  6<»    4. 

1860-1869 

—  o.  44  5  "> 

—  o.  <u  69 

V.  \J^   ^ 
—  o.  35  1  6 

1870-1880 

—  o.  52  57 

jj  _•? 
—  i.  25  67 

—  o.  30  24 

1881-1892 

—o.  84  80 

—  o.  73  102 

—  o.  37  26 

The  remarkable  feature  of  these  results  is  the  near  approach 
to  constancy  in  the  values  of  the  numbers  in  each  column, 
after  the  secular  variation  is  allowed  for,  and  the  large  magni- 
tude of  the  corrections.  The  most  natural  conclusion  is  that 
the  reduction  from  the  limb  of  the  planet  or  the  observed 
center  of  light  to  the  true  center  was  too  small  by  an  amount 
which,  at  the  mean  distance  of  the  Sun,  must  have  been  nearly 
or  quite  a  second  of  arc  (cf.  §  3).  The  adopted  semidiameter 
3".4  seems  so  well  established,  both  by  micrometric  measures 
and  by  heliometer  measures  during  transits  of  Mercury,  that 
such  a  correction  to  the  diameter  seems  inadmissible. 

I  have  not  yet  been  able  to  enter  upon  the  investigation  of 
the  source  of  this  anomaly.  A  very  important  question  is  that 
of  its  influence  on  the  results.  Since  a  constant  error  in  the 
radius  vector  of  a  planet  would  have  opposite  effects  on  the 
elements  in  different  points  of  the  relative  orbit,  it  may  be 
inferred  that  the  effect  of  the  error  would  be  nearly  eliminated 


33,  34]    COMPARISON  OF  OBSERVATIONS  OF  MERCURY.  69 

in  an  extensive  series  of  observations  distributed  equally 
between  the  two  elongations.  Actually,  however,  there  seems 
to  have  been  an  appreciable  lack  of  symmetry  in  this  respect, 
as  the  influence  of  the  unknown  quantity  upon  the  other 
unknowns  is  not  inconsiderable.  Although  the  law  of  change, 
as  shown  in  the  preceding  table,  does  not  correspond  to  the 
magnitude  of  the  coefficient  of  6r",  this  coefficient  being  rela- 
tively too  small  near  inferior  conjunction  and  too  large  near 
superior  conjunction,  it  is  still  probable  that  through  the  intro- 
duction and  elimination  of  dr"  a  large  part  of  the  injurious 
effect  is  eliminated. 

Comparison  of  transits  and  meridian  observations  of  Mercury. 

34.  Another  remarkable  result  which  may  be  associated  with 
this  is  shown  by  the  difference  between  the  solutions  A  and  B, 
in  the  case  of  the  eccentricity  and  perihelion  not  only  of  the 
planet,  but  of  the  Sun.  It  will  be  seen  that  the  meridian 
observations  alone  give  a  negative  correction  to  the  eccen- 
tricity of  the  planet,  while,  when  the  transits  are  included, 
the  correction  becomes  positive.  That  this  is  due  to  a  system- 
atic cause  running  through  the  observations  is  shown  by  the 
fact  that  the  same  thing  is  true  of  the  secular  variation  of 
the  eccentricity.  This  relation  of  the  correction  to  its  secular 
variations  holds  true  for  three  of  the  four  relative  elements, 
and  for  the  eccentricity  and  perihelion  both  of  the  planet  and 
of  the  Earth.  In  the  case  of  the  Earth's  perihelion,  however, 
there  is  a  nearer  approach  to  conformity  between  the  two 
results. 

There  is  yet  another  anomaly  in  this  connection,  which  indi- 
cates a  very  considerable  systematic  error  in  the  older  meridian 
observations,  which  is  not  completely  eliminated  from  the  ele- 
ments. If  we  take  the  values  of  the  unknown  quantities  and 
their  secular  variations,  which  result  from  the  two  solutions, 
and  substitute  them  in  the  linear  functions  of  the  corrections 
to  the  elements  derived  from  the  transits  alone,  namely 

V  =  1.487  dl  -  0.487  drr  -  1.137  de  -  1.01  dl"  +  1.19  e"dn" 

+  1.58  de" 

W  =  0.716  dl  +  0.284  dn  +  0.896  de  -  0.97  dl"  -  1.11  e"d7t" 

-  1.62  de" 


70  MERCURY,  VENUS,  AND  MARS.      [34,  35,  36 

we  find  the  following  results : 

From  meridian  observations    V  =  —  2".99  +  0".69T 
From  November  transits  —1  .69  —  2  .63  T 

From  combined  solution  —2  .77  —  2  .30  T 

From  meridian  observations  W  =  -f-  0".89  —  4".55  T 
From  May  transits  alone  +1  .39  +  1  .84  T 

From  combined  solution  +1  .39  +  0  .42  T 

We  conclude  that,  had  no  transits  ever  been  observed,  the 
errors  of  the  elements  and  their  secular  variations,  derived 
from  the  great  mass  of  meridian  observations,  would  have 
caused  an  error  of  some  5"  per  century  in  the  heliocentric 
place  of  the  planet  at  the  times  of  the  May  transits,  and  of 
some  3"  at  the  time  of  the  November  transits. 

The  fact  that  the  combined  solution  B  satisfies  the  transits 
so  much  better  than  A,  although  the  total  weight  of  equations 
A  is  so  much  greater  than  that  of  the  transit  equations,  shows 
that  the  meridian  observations  give  only  weak  results  for  the 
functions  in  question. 

Meridian  observations  of  Venus. 

35.  So  far  as  the  meridian  observations  are  concerned,  those 
of  Venus  were  treated  on  the  same  general  plan  as  the  observa- 
tions of  Mercury.    The  following  are  the  principal  points  of 
difference : 

1.  The  hypothetical  quantity   dr"  is  omitted.     Hence  no 
index  to  the  consistency  of  the  observations  at  different  points 
of  the  relative  orbit  can  be  derived  from  the  solution. 

2.  Tenths  of  a  unit  were  included  in  the  coefficients  of  the 
equations,  and  no  modification  was  made  in  the  units.    The 
units  and  tenths  were,  however,  dropped  in  the  final  solution 
of  the  normal  equations. 

Results  of  observed  transits  of  Venus. 

36.  We  put,  at  the  time  of  a  transit, 

v,  the  longitude  in  orbit  of  Venus ; 
Z,  its  mean  longitude,  or  the  mean  vame  of  v; 
fi,  A,  its  ecliptic  latitude  and  longitude; 
L,  the  Sun's  true  longitude. 


36  ]   EQUATIONS  OF  CONDITION  FROM  TRANSITS  OF  VENUS.     71 

Then 

tf  A  =  cos   i  6v  +  sin2  i  66 

=  0.99S2  dv  +  0.0592  sin  i  36 

We  thus  have,  for  the  dates  of  the  observed  transits, 

1761-'69 ;  dp  =  —  0.0592  6v  +  0.9982  sin  i  S6 
1874->82  ;£/?=  +  0.0592  Sv  -  0.9982  sin  *  dd 

I  have  discussed  very  fully  the  observations  of  the  transits 
of  1761  and  1769  in  Astronomical  Papers,  Vol.  n.  The  final 
results  which  I  shall  use  are  found  on  page  404  of  that  volume. 
Here  I  have  put. 

x,  correction  to  A  —  L ; 
—  y,  correction  to  /?, 

the  Sun's  latitude  being  supposed  to  require  no  correction. 
The  values  of  x  and  y  for  1769  are  distinguished  by  an  accent. 
I  have  also  represented  by  z2  and  23  the  corrections  to  the  dif- 
ference of  the  semidiameters  of  the  Sun  and  planet,  for  the 
respective  internal  contacts,  to  which  may  be  added  the.  un- 
known but  probably  nearly  constant  quantity  due  to  personal 
error  in  estimating  the  time  of  contact.  From  their  very  nature 
these  quantities  do  not  admit  of  accurate  determination,  and 
must  therefore  be  eliminated  from  the  equations.  From  the 
observations  of  internal  contact  are  derived  the  following  four 
equations : 

1761  II;  —  ,87a?  +  .50  y  +  z2  =  -  0".07 
HI ;  +  .68  +  .73  +  23  =  -  0".06 

1769  II;  -  .64  a?7  -  .11  y'  +  z2  =  -  0".27 
III;  +  .84  -  .55  +  23  =  +  0".02 

We  have  here  more  unknown  quantities  than  equations,  so 
that  it  is  not  practicable  to  determine  them  all  separately. 
What  I  have  done  has  been  first  to  assume  ^  =  23.  This  pre- 
supposes that  the  distance  of  centers  at  the  estimated  appa- 


72  MERCURY,  VENUS,   AND  MARS.  [36 

rent  contact  at  egress  is,  in  the  general  mean,  the  same  as  at 
ingress.  The  result  of  any  error  in  this  hypothesis  will  be 
almost  completely  eliminated  from  the  mean  latitude  at  the 
two  transits,  but  not  from  the  longitude. 

Still,  the  values  of  x  and  y  can  not  be  separately  determined; 
I  have  therefore  so  combined  the  equations  as  to  obtain  mean 
values  of  x  and  y  for  the  two  contacts,  assuming  that  this 
would  be  the  result  of  supposing  these  quantities  to  have  the 
same,  values  at  both  epochs.  Calling  these  values  x"  and  y", 
we  have  by  addition  and  subtraction,  supposing  z2  =  23, 


-  0.39  a?"  +  2.55  y"  =  0".12 
3.03  #"4-  0.45  y"  =  0".30 


We  thus  have* 


x"  =  +  0".09 
y1'  =  +  0".06 

These  corrections  are  not  applicable  to  the  coordinates  from 
LEVERRTER'S  .  tables  as  they  stand,  but  to  those  quantities 
as  corrected  by  the  following  amounts  : 

A\  =  4-  0".25 
Afi=  4-2".00 

*  In  a  second  approximation  to  these  quantities,  which  may  be  made 
after  the  correction  to  the  «enteimial  motion  of  the  node  is  determined, 
we  should  put,  on  account  of  this  correction, 


The  solution  would  then  give 


I  have  carried  through  a  more  careful  approximation  in  a  subsequent 
chapter. 


36]  EQUATIONS  OF  CONDITION  FROM  TRANSITS  OF  YENUS.     73 

We  thus  find,  for  the  corrections  to  LEVERRIER'S  tables  at 
the  epoch  1765.5, 

d  A  -  £L  =  +  0".09  +  0".25  =  +  0".34 
3  =  _  0".06      2".00  .=  +  1".94 


and  hence 

dv  =  +  0".22  +  0".998  £L 
sin  i  6  S  =  +  1".95  +  0//.059  6L 


A  still  farther  modification  is  required  to  the  tabular  longi- 
tude on  account  of  the  correction  to  the  mass  of  the  Earth 
used  by  LEVERRIER,  and  hence  to  the  periodic  perturbations 
in  longitude.  This  correction  is  +  0".20.  We  thus  have  for 
the  correction  to  the  orbit  longitude  of  Venus— 

dv  =  +  0".02  4-  0".998  6  L 

For  the  results  of  the  transits  of  1874  and  1882  I  have 
depended  entirely  on  the  heliometer  measures  and  photo- 
graphs made  by  the  German  and  American  expeditions, 
respectively.  The  definitive  results  of  the  German  observa- 
tions, as  worked  up  by  Dr.  AUWERS,  are  found  in  Vol.  V  of 
the  German  Keports  on  the  Transits.*  The  American  photo- 
graphic measures  of  1874  have  not  been  officially  worked  up 
and  published,  but  a  preliminary  investigation  from  the  data 
contained  in  the  published  measures  was  made  by  D.  P.  TODD, 
and  published  in  the  American  Journal  of  Science,  Vol.  21, 
1881,  page  491.  The  measures  of  1882  have  been  definitively 
worked  up  by  HARKNESS,  but  only  the  results  published. 
They  are  found  in  the  report  of  the  Superintendent  of  the 
TJ.  S.  Naval  Observatory  for  the  year  1890. 

The  corrections  to  the  geocentric  Eight  Ascension  and 
Declination  of  Venus  relative  to  the  Sun  thus  derived  are 

*  Die  Venus-durchgiinge  1874  und  1882  Bericht  uber  die  Deutsclien 
Beobachtungen  Fuufter  Band,  Berlin,  1893. 


74  MERCURY,   VENUS,   AND   MARS.  [36 

given  in  the  following  table.  In  taking  the  mean  the  weights 
are  not  strictly  those  which  would  result  from  the  probable 
errors  as  assigned,  but,  in  accordance  with  a  general  princi- 
ple, independent  results  have  received  a  weight  more  near  to 
equality  than  would  be  indicated  by  the  mean  errors. 

1874:  German,         6  E.  A.  =  +  4.77  ±  0.28 
American,       .      .      .     +  4.14  ±  0.30 

Adopted,    .      .      .     +4.44 

German,        6  Dec.  =  +  2.28  +  0.10 
American,      .      .      .     -f  2.50  ±  0.30 

Adopted,    .      .      .     +2.34 

1882:  German,         6  E.  A.  =  +  9.03  +  0.12 
American,      .      .      .     +  9.10  +  0.08 

Adopted,    .      .      .     +9.07 

German,         d  Dec.  =  +  2.02  i  0.06 
American,       .      .      .     +2.02  +  0.08 

Adopted,    .      .      .     +2.02 

We  change  these  results  successively  to  geocentric  longi- 
tude and  latitude,  heliocentric  longitude  and  latitude,  and 
orbital  longitude  and  latitude.  The  results  of  these  several 
changes  are  as  follow: 


Corr.  in  geoc.  long. 
Corr.  in  lat. 

1874. 
+  3''.853 

+  2  .724 

1882. 
+  8".077 

+  2.  971 

Corr.  in  hel.  long. 
Corr.  in  hel.  lat. 

-1  .415 
+  1  .001 

-2  .965 
+  1  .091 

Corr.  in  orbital  long. 
Value  of  sin  i  6  0 

-1  .35 
-1  .08 

-2  .90 

-1  .26 

37]  EQUATIONS  FROM  TRANSITS   OF  VENUS.  75 

Equations  from  transits  of  Venus. 

37.  The  corrections  to  the  heliocentric  positions  of  Yenus 
and  the  Earth,  as  thus  found,  are  now  to  be  expressed  in 
terms  of  corrections  to  the  elements.  The  results  of  this 
expression  are  shown  in  the  following  equations: 

Equations  given  by  the  corrections  to  the  orbital  longitude. 

I.  Epoch,  1765.5;  T=  —  3.90  ;  weight  =  200 

0.992  61  +  1.17  eSn  +  1.62  de  -  0.976  61"—  1.81  e"dn"-  0.85  de" 

=  +0".02±  O."15 

II.  Epoch,  1874.9  ;  r  =  +  0.48;  weight  =  400 

-  0".88/*  +  1.009  61  -  1.223  edn  -   1.596  6e  -  1.030  61" 
$*"  +  0.817  6e"  =  -  1".35  ±  0".08 


III.  Epoch,  1882.9;  T  =  +  0.80;  weight  =  800 

0".60,w+  1.008  61  -  1.146  edn  -  1.651  6e  -  1.028  <M"+  1.825  6"$w" 

+  0.900  $6"  =  —  2".90  i  0.'/027 

Equations  given  by  the  corrections  to  the  orbital  latitude. 


I.  1765.5;  sin  i66-  0.057  <«"-  0.11  ^^^"-O.OS^^  +  1".95 

i  0".10 


II.  1874.9;  sinid«9-0. 

±  0/7.04 


III.  1882.9;  sin^^-0. 

i  0".019 

The  weights  assigned  to  these  three  equations  are,  respec- 
tively, 200,  600,  and  1,600. 

Before  using  these  equations  the  corrections  to  the  elements 
were  transformed  into  the  unknown  quantities  denned  in  §27, 
and  their  secular  variations  by  multiplying  the  coefficients  by 
the  factors  given  on  page  56. 


76 


MERCURY,  VENUS,  AND  MARS. 


[38,  39 


Solutions  of  the  equations  for  Venus. 

38.  The  parts  of  the  normal  equations  formed  from  the 
preceding  conditional  equations  were  added  to  the  parts  from 
the  meridian  observations,  and  the  resulting  solution  B 
obtained.  As  in  the  case  of  Mercury,  a  solution  A  was  made 
of  the  normal  equations  derived  from  the  meridian  observa- 
tions alone.  The  results  are  as  follows : 

VENUS. 
Results  of  solutions  of  the  normal  equations. 


Unknowns. 

Factors. 

Corrections  of  elements. 

Symbol. 

A. 

B. 

Symbol. 

A. 

B. 

[>] 

—o.  0708 

—o.  0834 

7- 

<5  m  :  m 

—  o.  496 

—0.^584 

'  /  ' 

—o.  1435 

—o.  1501 

5- 

61 

-0.718 

—0-751 

"  J  ' 

+o.  1156 

+o.  1340 

6. 

6  T 

+o.  694 

+o.  804 

N; 

4-o.  0164 

-j-o.  0106 

7- 

sinJrfN 

+o.  115 

+o.  074 

e 

+o.  0941 

-j-o.  1003 

3- 

fit 

+o.  282 

-|-o.  301 

7T  ] 

4-o.  0628 

-j-o.  0764 

3- 

e6K 

4-o.  1  88 

+o.  229 

e 

4-o.  0246 

+0.0271 

4- 

6e 

-j-o.  098 

4-o.  1  08 

~  e"\ 

4-o.  0336 

4-o.  0318 

2-5 

5e" 

4-o.  084 

+o.  080 

~ir"' 

—o.  0274 

—0.0212 

2. 

e"tv" 

—o.  055 

—o.  042 

'  a 

+o.  4742 

+o.  4642 

I. 

a 

+o.  474 

+o.  464 

i  6  ''• 

—o.  0383 

-o.  0375 

5- 

6 

—  o.  192 

—o.  1  88 

*///• 

—  o.  0768 

—o.  0743 

4- 

61" 

—o.  307 

-o.  297 

'  /  ' 

—  o.  1846 

-o.  1983 

20. 

Dtd/ 

—3.  692 

—3.  966 

:*: 

+o.  0970 
—  o.  0561 

+o.  1088 
—o.  0594 

24. 

28. 

DtJ 
sinJDtN 

+2.  328 
—I-57I 

+2.611 
-1.663 

e 

+o.  1472 

+o.  1644 

12. 

Dt  e 

+  1.766 

+  r-973 

TT 

+0.0555 

4-o.  0698 

12. 

^Dt7r 

+o.  666 

+o.  838 

£ 

-(-o.  0182 

-j-O.  O2O2 

1  6. 

Dte 

+o.  291 

+o.  323 

"e"\ 

4-o.  0283 

+0.0317 

10. 

Dt^x/ 

+o.  283 

7T// 

+o.  0399 

+o.  0506 

8. 

e"  Dt  TT// 

+o.  3J9 

4-o.  405 

a 

—  o.  0820 

-o.  0347 

4- 

Dta 

—o.  328 

—o.  139 

!  <*  s 

—0.  0020 

—  O.  OOO2 

20. 

Dt  d 

—o.  040 

—o.  004 

L///- 

--o.  0562 

—o.  0662 

1  6. 

EM/" 

—o.  899 

—  i.  059 

Mean  epoch  of  correction,  1863.0 

Comparison  of  transits  of  Venus  with  meridian  observations. 

39.  To  show  to  what  extent  the  results  of  the  meridian 
observations  differ  from  those  of  the  observed  transits  over 
the  Sun,  we  form  the  values  of  the  absolute  terms  of  the 
equations  of  condition,  §37,  first  by  substituting  the  values 
A  of  the  corrections,  and  then  the  values  B.  We  thus  have : 


39,  40]          EQUATIONS  FROX  TRANSITS   OF  VENUS.  77 

Residuals  in  orbital  longitude. 

1765.5.  1874.9.          1882.0. 

(a)  From  meridian  obs.  alone  .      -  0".07  -  1".36  -  2".54 

( ft)  From  combined  solution     .     +  0".04  -  1".43  -  2".78 

(y)  From  transits  alone  .     .     .     +  0".02  -  I". 35  -  2".90 

Discordance,  (y)-(tf)    .     .      +0".09  +  0".01  -  0".36 

Discordance,  (y)- (ft)     .     .     -  0".04  +    0.08  -0".12 

Residuals  in  orbital  latitude. 

1765.5.  1874.9.          1882.9. 

(a)  From  meridian  obs.  alone  .     +  1".92  -  0".77  -  0".96 

(ft)  From  combined  solution     .     +  2".06  -  0".91  -  1".12 

(y)  From  transits  alone  .     .     .     +  1".95  -  1".08  -  1".26 

Discordance,  (y)- («)    •     •     +  0".03  -  0".31  -  0".30 

Discordance,  (y)  —  (ft)    .     .     -  0".ll  -  0".17  -  0".14 

It  will  be  seen  that  the  combined  solution  represents  the 
observations  of  the  transits  much  better  here  than  in  the  case 
of  Mercury. 

Solution  of  the  equations  for  Mars. 

40.  As  the  formation  of  the  normal  equations  for  Mars  was 
approaching  its  end,  a  singular  discordance  among  the  resid- 
uals of  the  partial  normal  equations  for  different  periods  was 
noticed.  On  tracing  the  matter  out  it  appeared  that  while  the 
correction  of  the  geocentric  longitude  of  LEVERRIER'S  tables 
in  1845  and  again  in  1892  was  quite  small,  the  correction  in 
1862  was  considerable.  Now  there  is  an  inequality  of  long 
period,  about  forty  years,  in  the  mean  motion  of  Mars,  depend- 
ing on  the  action  of  the  Earth,  and  having  for  its  argument 
15#7  —  Sg.  This  coefficient  is  of  the  seventh  order  in  the  eccen- 
tricities, and  the  terms  of  the  ninth  or  even  of  the  eleventh 
order  might  be  sensible  in  a  development  in  powers  of  the 
eccentricities  and  sines  or  cosines  of  multiples  of  the  mean 
longitudes.  The  conclusion  which  I  reached  was  that  the  the- 
oretical value  of  this  coefficient  was  not  determined  with  suffi- 
cient precision.  As  the  work  of  solving  the  equations  could 
not  wait  for  a  new  determination  and  a  new  formation  of  the 
absolute  terms  of  the  normal  equations,  it  was  decided  to  make 
an  approximate  empirical  correction  to  the  theory.  This  was 
used  to  correct  the  absolute  terms  of  the  partial  normal  equa- 


78 


MERCURY,  VENUS,  AND  MARS. 


tions  for  each  period,  and  the  solution  was  then  proceeded 
with.  The  chances  seein  to  be  that  by  this  process  the  inju- 
rious effect  of  the  error  upon  the  elements  derived  from  the 
equations  would  be  inconsiderable;  this  is,  however,  a  point 
on  which  it  is  impossible  to  speak  with  certainty.  It  is  the 
intention  of  the  writer  to  recompute  the  doubtful  terms  of  the 
perturbations,  and,  if  possible,  reconstruct  the  absolute  terms 
of  the  normal  equations  in  accordance  with  the  corrected 
theory.  Meanwhile,  the  present  work  necessarily  rests  on  the 
imperfect  theory  with  the  approximate  empirical  corrections, 
which  are  as  follow : 

M  =  0".30  cos  (150'  -  8-7  -  223°) 
edn  =  0".15  cos  (150'  —  80) 

As  the  elements  of  Mars  are  derived  wholly  from  meridian 
observations,  only  one  set  of  equations  of  condition  was  formed. 
The  results  of  the  solution  are  shown  in  the  following  table : 

MARS. 


Unknowns. 

Factors. 

Corrections  of  elements. 

Symbol. 

Value. 

Symbol. 

Value. 

[  tn/~\ 

—  .  02278 

0-3 

6  m  :  m 

-o.  007 

1  ] 

—  .  44854 

2. 

61 

-o.  897 

N; 

+•  05479 
+.06724 

2-5 
2-5 

SinJJN 

+o.  137 
+o.  1  68 

e 

+  .43803 

•V 

6  e 

+o.  626 

TT 

—  .  05056 

1^-Q 

6w 

—o.  722 

e 

+.07474 

4- 

6s 

+o.  299 

e"\ 

—  .  49898 

2. 

be" 

—  o.  998 

V/x" 

—  .42409 

2. 

e'f  6  K" 

—o.  848 

a 

+  •  18545 

5- 

a 

+o.  927 

;  6  ' 

—  .  04536 

5- 

6 

—  o.  227 

• 

///' 

+  .05786 

3- 

61" 

+o.  174 

"  /  " 

+.  16605 

8. 

Dtd/ 

+1.328 

;  JT 

+  •  13408 
—  .  02263 

10. 

10. 

DtJ 
SinJDtN 

+1.341 

—o.  226 

e 

—  .  03180 

V1 

Dt<? 

—o.  182 

TC 

—  .  00928 

A^Q 

Dt7T 

-o.  530 

e 

+.  06097 

1  6. 

Dt  £ 

+0.976 

'  £// 

—  .  12597 

8. 

l}^e// 

—1.  008 

"TT//= 

+.00853 

8. 

e"  Dt  TT" 

+o.  068 

a 

—  .  09670 

20. 

Dta 

—  i.  934 

"  6  ] 

—.01168 

20. 

Dt^ 

-o.  234 

\l>'\ 

+.  13111 

12. 

Dtrf/" 

+1.573 

41]  REFERENCE   TO  THE  ECLIPTIC.  79 

Reference  to  the  ecliptic. 

41.  In  all  the  preceding  determinations  the  planes  of  the 
orbits  are  referred  to  the  plane  of  the  Earth's  equator,  or,  to 
speak  more  exactly,  to  a  plane  through  the  Sun  parallel  to  the 
Earth's  equator.  As  in  astronomical  practice  the  ecliptic  is 
taken  as  the  fundamental  plane,  it  is  necessary  to  investigate 
the  reduction  of  the  elements  from  one  plane  to  the  other. 

Let  us  consider  the  spherical  triangle  formed  on  the  celestial 
sphere  by  the  plane  of  the  orbit,  the  plane  of  the  ecliptic,  and 
the  plane  of  the  Earth's  equator.  For  the  sides  and  opposite 
angles  of  this  triangle  we  have 

Sides:  N  6  y  • 

Opposite  angles  :  i  180°  —  J  e 

When  equatorial  coordinates  are  used,  the  position  of  the 
planet  is  considered  as  a  function  of  the  three  quantities 

N;         J;         e  (a) 

When  ecliptic  coordinates  are  used,  the  three  corresponding 
quantities  are 

0;         I-,          e  (b) 

Taking  the  set  of  quantities  (a]  as  the  fundamental  parts  of 
the  triangle,  and  expressing  the  corrections  of  the  other  parts 
as  functions  of  them,  we  have 


6  i  =  +  cos  rp6J  +  sin  ip  sin  JtfN  —  cos  Ode 
sin  i  d  9  =  —  sin  fidJ  +  cos  >/>  sin  J6N  +  cos  *  sin  Ode 

Taking  (b)  as  the  fundamental  parts,  we  have  for  the  correc- 
tions to  N  and  J 


d  J  =  cos  fi6i  —  sin  ip  sin  i$6  +  cos 
sin  J#N=  sin     6i  +  cos     sin  idd  —  cos  J  sin  Ntfs 


The  numerical  values  assigned  to  the  coefficients  in  these 
equations  are  those  corresponding  to  the  mean  epoch  1850. 
The  fact  that  they  change  somewhat  in  the  course  of  a  hundred 
years  has  not  been  taken  account  of.  The  future  astronomer 
will  meet  with  a  real  difficulty  in  that  the  corrections  to  a 


80  MERCURY,   VENUS,   AND  MARS.  [41 

set  of  elements  at  one  epoch  do  not  accurately  correspond  to 
similar  corrections  at  another  epoch.  It  is  impossible  to  do 
away  rigorously  with  the  difficulty  thus  arising,  except  by 
introducing  a  more  general  system  of  elements  than  elliptic 
ones.  The  error  is,  happily,  not  important  in  the  present  state 
of  astronomy.  The  equations  in  question  for  the  three  planets 
are  as  follow: 

Mercury. 

di  =  +  .799  (5  J  +  .602  sin  J  fi  X  —  .688  fo 
sin  idO  =  -  .602  6J  +  .799  sin  J  d  N  +  .721  fa 

Venus. 


di  =  +  .373  6  J  +  .928  sin  J  £N  -  .255  fa 
sin  idS  —  —  .928  d  J  +  .373  sin  J  d  N  4-  .967  de 

Mars. 

di  =      .703  6  J  +  .712  sin  J  tf  N  -  .664  6s 
sin  id  6  =  -  .712  3J  +  .703  sin  J  3X  +  .747  fa 


For  the  inverse  relations  we  have  — 

Mercury. 

3J=  .799  <H  -  .602  sin  idti  +  .983  de 
sin  J  SIS  =  .602  6i  +  .799  sin  id  6  -  .162  de 

Venus. 

6  J  =  .373  di  —  .928  sin  idO  +  .990  de 
sin  J  d  N  =  .928  di  +  .373  sin  idB  -  .125  fa 

Mars. 

6  J  =  .703  <M  -  .712  sin  id  6  +  .998  fa 
sin  J  £N  =  .712  <«  +  .703  sin  id  B  -  .052  fa 


CHAPTER  IY. 

COMBINATION  OF  THE  PRECEDING  RESULTS  TO  OBTAIN 
THE  MOST  PROBABLE  VALUES  OF  THE  ELEMENTS 
AND  OF  THEIR  SECULAR  VARIATIONS  FROM  OBSER- 
VATIONS ALONE. 

In  the  two  preceding  chapters  are  derived  four  separate 
values  of  the  six  corrections,  <*,  #,  6e,  61",  de",  and  e"6n",  and 
of  their  secular  variations,  which  pertain  to  the  orbit  and 
motion  of  the  Earth  relative  to  the  stars.  We  have  now  to 
combine  these  four  results  so  as  to  derive  the  most  probable 
values  of  the  twelve  unknown  quantities  in  question. 

Deviations  from  the  method  of  least  squares. 

42.  If  we  applied  without  modification  the  principles  of  the 
method  of  least  squares,  we  should  first  eliminate  the  elements 
and  secular  variations  for  each  planet  from  the  normal  equa- 
tions given  by  observations  of  that  planet,  which  would  leave 
us  with  three  sets  of  normal  equations,  containing  only  the 
twelve  quantities  depending  on  the  motion  of  the  Earth.  We 
should  then  reduce  these  normal  equations  to  equality  of 
weight,  by  multiplying  each  of  them  by  the  appropriate 
factors,  and  we  should  then  consider  the  observed  corrections 
to  the  solar  elements  derived  from  observations  of  the  Sun 
alone  as  affording  equations  of  condition  to  be  reduced  to  the 
adopted  system  of  weights,  and  then  multiplied  by  their  coeffi- 
cients and  added  to  the  normal  equations.  The  solution  of 
the  single  set  of  normal  equations  thus  formed  would  lead  to 
the  definitive  values  of  the  solar  elements  and  of  their  secular 
variations,  which,  being  substituted  in  the  eliminating  equa- 
tions from  each  planet,  would  lead  to  the  definitive  elements 
of  the  planet  and  of  their  secular  variations. 

This  proceeding  is  not,  however,  advisable  in  the  present 
case,  because,  owing  to  the  immense  mass  of  material  worked 

5690  N  ALM 6 

81 


82  PROBABILITY   OF    ERRORS.  [42 

up,  the  errors  to  be  principally  feared  are  not  the  accidental 
ones,  of  which  alone  the  method  of  least  squares  takes  account, 
but  the  systematic  ones  arising  principally  from  personal 
equation  and  imperfect  reduction  of  the  observations  to  the 
actual  center  of  the  planet  or  of  the  Sun.  These  errors  affect 
different  elements  in  very  different  ways  and  to  different 
amounts;  from  some  they  will  be  almost  completely  elim- 
inated and  from  others  they  will  not.  We  must  therefore  pro- 
ceed by  a  tentative  process,  ascertaining  at  each  step,  so  far 
as  possible,  how  each  result  .will  come  out  before  we  accept  it 
as  final,  to  be  combined  with  other  results.  In  doing  this  it 
is  necessary  to  deviate  so  widely  from  what  are  commonly 
regarded  as  fundamental  principles  of  the  theory  of  the  com- 
bination of  observations  that  a  brief  presentation  of  the  prin- 
ciples involved  is  appropriate. 

It  is  frequently  accepted  as  an  axiom  that  when  we  have 
several  non-accordant  determinations  of  the  same  quantity, 
between  which  we  have  no  reason  for  choosing,  the  most  prob- 
able value  is  the  arithmetical  mean.  The  operation  of  taking 
the  arithmetical  mean  is,  in  fact,  the  simplest  application  of 
the  method  of  least  squares.  The  fundamental  hypothesis  on 
which  this  method  rests  is  that  the  probability  of  an  error  of 
magnitude  i  x  is  given  by  the  well-known  exponential  equa- 
tion 

h    —M** 
<p  (Ji,  x)dx  =  —  —e        dx  (a) 


h,  the  modulus  of  precision,  being  a  constant.  It  was  shown  by 
GAUSS  that  this  function  for  the  probability  follows  rigorously 
from  the  principle  of  the  arithmetical  mean.  It  therefore  fol- 
lows that  the  method  of  the  arithmetical  mean,  and  therefore 
that  of  least  squares,  is  rigorously  correct  only  so  far  as  the 
law  of  error  is  expressed  by  the  above  exponential  function. 

It  scarcely  needs  to  be  pointed  out  that,  as  a  matter  of  fact, 
the  law  of  error  in  question  is  not  true.  Not  only  so,  but  in 
astronomical  experience  it  deviates  from  the  truth  in  a  way 
admitting  of  precise  statement.  It  presupposes  that  the  mod- 
ulus of  precision  is  a  determinate  quantity.  Were  this  the 
case,  then,  to  take  a  single  instance,  the  probability  of  an 


42]  PROBABLE   ERRORS   AND   WEIGHTS.  83 

error  five  times  as  great  as  the  probable  error  would  be  less 
than  0.001,  and  the  probability  of  an  error  six  times  as  great 
would  be  about  0.0001.  This  is  not  true,  because,  taking  the 
function  q>  (ft,  x)  as  a  basis,  we  may  say  that  the  modulus  of 
precision,  ft,  is  nearly  always  in  practice  an  uncertain  quan- 
tity. Let  us  then  put 

hi,  lit,  h3,     .    .    . 
for  the  possible  values  of  ft,  and 


for  the  several  probabilities  that  h  has  these  respective  values. 
Then  the  probability  function  will  become 

<p(x)=Pi  <P(h\,  x)  +Pz(p'(h*,  oo)  +     ...  (6) 

Now  this  form  can  not  be  reduced  to  the  form  (a)  with  any 
value  whatever  of  the  modulus  h.  If  we  make  the  closest  rep- 
resentation possible,  we  shall  have  a  curve  in  which  small 
values  and  large  values  of  x  are  relatively  less  probable  as 
compared  with  the  facts  than  are  intermediate  values.  To 
show  that  this  is  the  actual  case,  let  us  suppose  that  we  have 
three  determinations  of  an  unknown  quantity.  If  we  proceed 
in  the  usual  way,  we  should  infer  the  value  of  ft,  the  measure 
of  precision,  from  the  discordance  of  these  three  values.  But 
it  is  evident  that  this  determination  of  ft  w.ould  be  very  uncer- 
tain. Should  the  three  values  chance  to  be  fortunately  accord- 
ant, then,  proceeding  in  the  usual  way,  our  function  would  lead 
to  the  conclusion  that  the  probability  of  an  error  of  a  certain 
magnitude  in  the  mean  was  very  small,  when,  as  a  matter  of 
fact,  it  might  be  very  considerable.*  The  value  of  ft  being 

*  To  take  a  simple  and  quite  possible  instance,  let  three  observations  of 
a  star  with  a  meridian  circle  give,  for  the  seconds  of  declination,  0".4,  0".5, 
and  0".6.  By  the  canons  of  least  squares  the  mean  result  would  be 

0".50  ±  0".039 
and  the  probability  of  an  error  as  great  as  0".l  would  come  out  about  0.08. 


84  PROBABLE  ERRORS  AND   WEIGHTS.  [42 

uncertain,  the  true  form  of  the  function  is  not  (a}  but  (b).  It 
follows  that  we  may  lay  down  the  following  general  rule  : 

The  best  value  from  a  system  of  non- accordant  determinations 
is  not  the  arithmetical  mean,  but  a  mean  in  which  less  iceiglit  is 
assigned  to  those  results  which  deviate  most  widely  from  the 
mean  of  the  others. 

I  have  considered  the  subject  from  this  point  of  view  in  the 
American  Journal  of  Mathematics,  Vol.  VIII,  p.  343,  and  given 
tables  for  determining  the  weights  to  be  assigned  to  the  results 
when  the  law  of  error  is  that  derived  from  several  hundred 
observed  contacts  of  the  limb  of  Mercury  with  that  of  the  Sun 
during  transits  of  the  planet. 

Another  well-known  defect  in  the  method  of  least  squares 
is  that  it  does  not  take  any  account  of  systematic  errors.  The 
greater  the  number  of  observations  that  are  combined,  the 
larger  the  proportion  in  which  the  errors  of  the  results  may 
be  due  to  the  systematic  errors  in  the  observations  or  the 
elements  of  reduction.  Although  such  errors  may  elude  inves- 
tigation so  far  as  their  determination  and  elimination  is  con- 
cerned, we  may  yet  be  able  to  point  out  their  origin,  and  to 
show  to  what  extent  they  would  influence  each  separate  result. 
Of  some  results  we  can  say  with  entire  confidence  that  they 
are  but  slightly  affected  with  systematic  error  5  of  others,  that 
they  may  be  very  largely  so  affected.  In  the  latter  case,  the 
weights  of  the  results,  as  determined  from  the  solution  of  the 
normal  equations,  give  no  clue  whatever  to  the  probable  mag- 
nitude of  the  error. 

The  result  of  this  is  that  in  the  following  paper  we  are  more 
than  once  confronted  with  the  following  problem:  Among 
several  determinations  of  a  quantity  one  is  known  to  be  free 
from  systematic  error  and  to  be  affected  with  a  well  determined 
probable  mean  error,  i  e.  There  are  also  one  or  more  other 
determinations  of  which  the  probable  error  is  unknown  and 
can  not  be  determined,  because  we  have  no  sufficient  knowl- 
edge of  the  probable  effect  of  systematic  errors  upon  the  result. 
What  shall  be  the  relative  weight  assigned  to  two  such  results 
in  order  to  obtain  the  mean?  The  decision  of  this  question  is 
necessarily  a  matter  of  judgment,  the  grounds  for  which  it 
might  be  extremely  prolix  to  state  at  length.  An  attempt  has 


42 1  PROBABLE   ERRORS  AND  WEIGHTS.  85 

been  made  in  these  cases  to  classify  the  results,  so  as  to  give 
a  general  idea  of  what  is  likely  to  be  their  modulus  of  pre- 
cision, and  weight  them  accordingly. 

Any  attempt  at  numerical  accuracy  in  such  an  estimate 
would  be  labor  thrown  away.  It  has  therefore  been  considered 
sufficient  in  such  cases  to  state  what  the  conclusion  of  the 
author  is,  leaving  its  revision  and  criticism  to  the  future 
investigator.  Indeed,  in  some  cases,  as  in  that  of  the  correc- 
tion to  the  centennial  motion  of  the  Sun  in  longitude,  a  con- 
venient round  number  has  been  chosen,  very  near  to  the  result 
of  well-decermined  weight. 

We  should  be  carrying  the  preceding  conclusions  too  far  if 
they  led  us  to  a  general  distrust  of  the  conclusions  reached  by 
the  method  of  least  squares.  The  doctrines  that  there  is  a 
necessary  limit  to  the  accuracy  with  which  astronomical  deter- 
minations can  be  made;  that  systematic  errors  necessarily 
affect  every  such  determination  5  and  that  the  canons  of  least 
squares  necessarily  lead  to  illusory  probable  errors,  are  too 
sweeping.  We  may  lay  down  the  general  rule  that  if  we  have 
a  sufficient  number  of  really  independent  determinations  of  an 
unknown  quantity,  of  which  we  individually  know  nothing 
except  that  they  are  the  results  of  actual  measures,  and  not 
mere  guesses,  then  the  arithmetical  mean  will  be  a  definite 
result,  the  probable  deviation  from  which  will  actually  follow 
the  law  given  by  the  canons  in  question  with  a  closeness 
which  will  continually  increase  with  the  number  of  independent 
determinations. 

If  we  have  such  knowledge  of  the  relative  values  of  the 
various  determinations  as  to  assign  greater  weight  to  some 
than  to  others,  the  result  will  be  still  better  when  those 
weights  are  used,  provided  always  that  they  are  assigned 
without  undue  bias  in  favor  of  those  results  which  most  nearly 
approach  the  value  supposed  to  be  approximately  correct. 

These  considerations  lead  me  to  a  policy  which  I  have 
always  adopted  when  it  was  easy  to  do  so  in  the  following 
discussions,  namely,  that  of  so  conducting  the  work  as  to 
lead  to  as  many  independent  determinations  of  a  quantity 
as  possible,  arid  of  always  giving  a  less  relative  weight  to  such 
sets  of  determinations  as  might  from  any  cause  whatever  be 


86  ELEMENTS   OF  EARTH'S   ORBIT.  [42,43 

supposed  affected  by  au  important  common  source  of  error. 
Where  the  independent  determinations  are  few  in  number,  the 
computation  of  a  definite  probable  error  is  impracticable,  and 
the  probable  mean  error  assigned  is  necessarily  a  result  of  a 
judgment  based  on  all  the  circumstances. 

Relative  precision  of  the  tico  methods  of  determining  the  elements 
of  the  Earth's  orbit. 

43.  When  the  system  of  determining  the  solar  elements  from 
observations  of  the  planets  as  well  as  of  the  Sun  was  originally 
decided  upon,  it  was  supposed  that  the  two  methods  would 
give  results  not  greatly  differing  in  accuracy  in  the  case  of  any 
of  the  elements.  This,  however,  is  proved  by  the  results  not  to 
be  the  case.  Attention  has  already  been  called  to  the  extreme 
consistency  of  the  values  found  for  the  correction  to  the  eccen- 
tricity and  perihelion  of  the  Earth's  orbit  from  observations  of 
the  Sun.  This  consistency  inspires  us  with  confidence  that 
the  probable  errors  of  the  corrections  to  the  elements  as  given 
do  not  exceed  a  few  hundredths  of  a  second.  But  the  deter- 
mination of  these  elements  from  observations  of  Mercury  and 
Venus  may  be  seriously  affected  by  the  form  of  the  visible 
disks  of  those  planets,  which  results  in  observations  being 
made  only  upon  one  limb  when  east  of  the  Sun  and  the  other 
limb  when  west  of  it.  Thus  personal  equation  and  the  uncer- 
tainty of  the  semidiameter  to  be  applied  in  each  case  may  have 
an  effect  upon  the  result.  But  personal  equation  is  likely  to  be 
smaller  in  the  case  of  Mercury  than  in  that  of  Venus,  owing 
to  the  smalluess  of  its  disk. 

There  is  another  circumstance  which  weakens  the  inde- 
pendent determination  of  the  Earth's  eccentricity  and  perigee 
from  observations  of  the  planets.  If  we  define  the  orbit  of  a 
planet,  not  as  a  curve,  but  as  the  totality  of  points  which  the 
planet  occupies  at  a  great  number  of  given  equidistant  moments 
during  its  revolution,  then  it  is  easy  to  see  that  the  general 
mean  effect  of  an  increase  of  the  eccentricity  is  to  displace  the 
entire  or.bit  toward  the  point  of  the  celestial  sphere  marked  by 
its  aphelion,  while  the  effect  of  a  change  of  its  perihelion  is  to 
move  the  entire  orbit  in  its  own  plane  in  a  direction  at  right 
angles  to  the  line  of  apsides.  The  result  is  that  in  a  series  of 


43,  44J  SECULAR  VARIATIONS  OF  THE  SOLAR  ELEMENTS.         87 

observations  of  a  planet  from  the  Earth  the  corrections  to  the 
eccentricity  and  perihelia  of  the  two  orbits  can  not  be  entirely 
independent,  and  we  can  determine  with  entire  precision  only 
two  linear  functions  expressive  of  the  relative  displacements 
just  described.  It  may  be  admitted  that,  were  observations 
exactly  similar  in  kind  made  around  the  entire  relative  orbit 
in  equal  numbers,  the  effect  of  the  principle  systematic  errors 
would  be  nearly  eliminated  from  the  result.  But  we  can  not 
rely  upon  this  being  the  case,  and  even  were  it  the  case  there 
would  probably  be  a  residual  effect  which  would  be  large  in 
proportion  to  the  interdependence  of  the  two  sets  of  correc- 
tions. But  in  this  connection  the  important  remark  is  to  be 
made  that,  so  far  as  these  systematic  errors  are  invariable, 
they  would  not  affect  the  secular  variations,  but  only  the  abso- 
lute values  of  the  elements.  We  may  therefore  assign  greater 
relative  weights  to  the  former  than  to  the  latter. 

So  far  as  we  cau  classify  the  results,  I  have  concluded  that 
in  the  case  of  the  secular  variations  of  f,  e"<  and  TI" ,  the  weight 
of  the  determination  from  Mercury  and  Venus  might  receive  a 
weight  one-fifth  that  from  the  Sun.  But  in  the  case  of  the 
absolute  values  of  these  quantities,  it  would  seem  from  the 
discordance  of  the  results  that  the  relative  weight  of  the 
planetary  results  should  be  much  smaller. 

In  dealing  with  the  common  error,  a,  of  the  adopted  Right 
Ascensions  of  the  stars,  it  is  to  be  remarked  that  we  may 
regard  the  observations  in  Eight  Ascension  as  fitted  to  give 
the  values  of  a  +  61",  while  61"  necessarily  depends  solely 
upon  the  observations  of  declination,  in  effect  if  not  in  form. 
Hence,  although  the  unknown  quantities  of  the  solution  are 
a  and  61",  I  have  deemed  it  best  to  derive  the  result  by 
regarding  a  +  61"  as  the  quantity  to  be  first  found,  instead  of 
a  itself. 

Secular  variations  of  the  solar  elements. 

44.  The  following  table  shows  the  corrections  to  the  tabular 
secular  variations  of  the  solar  elements,  as  they  have  been 
found  from  observations.  In  the  cases  of  Mercury  and  Venus 
the  results  of  both  solutions  are  given  for  the  sake  of  compari 
son,  although  only  solution  B  is  used.  The  relative  weights 


88 


ELEMENTS  OF  EARTH'S  ORBIT. 


[44 


have  been  determined  by  the  considerations  already  set  forth. 
In  the  case  of  Mars,  the  final  determinant  of  the  solution  for 
the  solar  elements  came  out  so-  nearly  evanescent  as  to  show 
that  no  reliable  values  could  be  obtained,  a  result  which  we 


Corrections  to  the  secular  variations  of  the  solar  elements  derived 
from  observations  only. 


Dt*« 

Dt<J/" 

Dt#f* 

From  observations  of  — 
The  Sun          

"      w. 

4-0.  48    5 

"      w. 

—  o.  97     i 

"      TC/. 
+°-  23    5 

Mercury,  solution  A  .  
"               "        B 

-4-0.27 

-4-O    3Q       I 

—0.58 
i  26     i 

—0.47 

1  O    32       I 

Venus,  solution      A  _  _ 

-f  o.  29 

—  o.  90 

4-0.28 

«          «            B 

4-O.  32       I 

—  i.  06     i 

4-o.  72    i 

Mars 

+  1.03    | 

Mean 

-1-0.48 

—  I.  IO 

4-o.  26 

Adopted  _  .   _     .      

+0.48 

.     —  I.OO 

4~O.  21 

*"Dtc57r" 

Dt(a  +  d/") 

Dta 

From  observations  of  — 
The  Sun 

"         IV. 

4-0.  32    5 

"      w. 

—0.  63      2 

// 

4-0.  34 

Mercury,  solution  A  

—  0.40 

-1.84 

—1.26 

«       B  

—  o.  29     i 

—  2.  05    3 

—  o.  79 

Venus,  solution      A 

4-o.  32 

O.  •?•? 

"          "           B 

4~o.  46     i 

—  I.  2O      2 

—  o.  14 

Mars 

Mean                 

-fo.  25 

—  I.  4O 

—  o.  30 

Adopted 

-l-o.  26 

—  I.  ^O 

—  o.  30 

might  expect,  because,  in  order  to  separate  the  principal  ele- 
ments of  the  Earth's  orbit  from  those  of  the  planet,  observa- 
tions should  be  continued  all  around  the  relative  orbit,  whereas, 
as  a  matter  of  fact,  they  are  generally  made  only  near  the  time 
of  opposition.  I  have  judged,  however,  that  the  correction  to 
the  secular  variation  of  the  obliquity  obtained  by  putting 
Dtdl"  =  —  1".00  MI  the  equation  for  Dttf  e  might  enter  with 
half  the  weight  that  it  does  in  the  cases  of  Mercury  and 
Venus.  Before  the  final  values  and  weights  of  the  quanti- 
ties in  the  table  had  been  definitively  revised,  provisional 
values  were  used  in  the  subsequent  part  of  the  investigation. 


44,  45]   CORRECTION  TO  THE  STANDARD  DECLINATIONS.  89 

These  provisional  values  are  given  in  the  last  line  of  the  table. 
Jt  is  also  to  be  noted  that  the  secular  variations  of  e,  e"  TT,  n" 
and  £  in  the  definitive  theory  and  tables  are  those  computed 
from  the  adopted  masses  of  the  planets. 

Correction  to  the  standard  of  Declination. 

45.  The  results  for  the  secular  variation  of  3,  the  common 
error  of  the  standard  Declinations  within  the  zodiacal  limits, 
are  not  given  in  the  table,  as  other  data  are  available  for  its 
determination.  The  following  shows  the  separate  values  of 
6  and  its  secular  variation,  derived  from  observations  of  the 
planets  to  Saturn  inclusive.  For  reasons  already  stated  obser- 
vations of  the  Sun  are  not  used  for  this  purpose. 

"          "  w. 

From  observations  of  Mercury,         6  =  —  0.18  -  0.49  T ;  2 

Venus,  -  0.19- 0.00  T;  1 

Mars,  -  0.21- 0.23 T;  4 

Jupiter,  -  0.04  -  0.43  T ;  3 

Saturn,  +  0.04-  0.68 T;  4 


Mean ;       d=—  0".09  —  0".42  T 
Adopted ;  <?  = -0  .08-0  .50T 

Not  only  observations  of  the  planets  but  those  of  the  fixed 
stars  are  available  for  the  determination  of  6  and  of  its  secular 
variation.  In  the  discussion  of  the  Declinations  derived  from 
observations  with  the  Greenwich  and  Washington  transit  cir- 
cles (Astronomical  Papers,  Vol.  II),  I  have  shown  that  the 
Greenwich  observations  indicate,  with  some  uncertainty,  a 
secular  variation  of  the  corrections  to  the  standard  declina- 
tions which  will  give  a  value  of  about  —  0".55  for  the  secu- 
lar variation  of  d.  But  BRADLEY'S  Declinations,  as  reduced 
by  AUWERS,  would  give  a  still  larger  negative  value,  approxi- 
mating to  an  entire  second.  As  the  value  which  we  may 
assume  for  d  does  not  greatly  influence  the  other  elements, 
I  have  adopted  as  a  convenient  probable  result,  the  varia- 
tion — 0".50  T. 


90  ELEMENTS  OF  EARTH'S  ORBIT.  [46 

Definitive  secular  variations  of  the  planetary  elements  from  obser- 
vations alone. 

46.  Having  decided  upon  the  adopted  values  of  the  six 
quantities  found  in  the  last  article,  we  regard  them  as  known 
quantities,  and  substitute  them  in  the  eliminating  equations, 
which  give  the  values  of  the  remaining  secular  variations. 
As  the  unknown  quantities  in  these  equations  are  not  the 
corrections  themselves,  but  certain  functions  of  them,  we  pre 
pare  the  following  table,  showing  the  formation  of  the  quan- 
tities which  are  to  be  substituted  in  the  several  equations. 
The  table  scarcely  seems  to  need  any  explanation,  except  that 
the  unknown  quantities  given  in  the  three  columns  on  the 
right  are  formed  by  dividing  the  secular  variations  for  twenty- 
five  years  by  the  coefficients  given  in  §  27. 

Adopted  secular  variations  of  the  solar  unknowns,  to  be  substi- 
tuted in  the  eliminating  equations  for  the  several  planets. 

Dt<M"  =  -T'.OO; 
Dttf  =—0  .50; 
Dta  =  -0.30; 

e"Dt<y7r"  =  +  0  .26; 
Dt6eff   =+0  .21;  [  e" 
Dtfo      =+0.48; 

To  facilitate  a  judgment  or  rediscussion  of  this  part  of  the 
process,  we  give  on  the  next  three  pages  the  normal  equations 
between  all  the  secular  variations  which  remain  after  the  cor- 
rections to  the  elements  of  the  Sun  and  planets  are  eliminated 
from  the  original  normal  equations.  We  give  these  rather 
than  the  eliminating  equations  "which  were  actually  used  in 
the  substitution,  because  they  show  more  fully  the  relations 
between  the  unknown  quantities,  and  can  therefore  be  better 
used  iu  any  ulterior  discussion.  Regarding  the  preceding  six 
quantities  as  known,  and  substituting  them  in  the  normal 
equations  for  the  secular  variations,  we  derive  the  definitive 
values  of  the  secular  variations  which  relate  to  the  planets. 
They  are  shown  in  the  next  table.  In  the  latter  the  values  of 
the  solar  elemen  ts  are  repeated  for  the  sake  of  completeness. 


Mercury. 

Venus. 

Mars. 

I" 

1'   =-0.250; 

-0.0625; 

-0.0833; 

d 

]'   =-0.125; 

-0.0250; 

-0.0250; 

a 

]'  =-0.075; 

-0.0750; 

-0.0150; 

n" 

]'  =  +  0.108; 

+  0.0325; 

+  0.0325; 

e" 

]'  =  +  0.087; 

+  0.0210; 

+  0.0262; 

£ 

]/  =  +0.120; 

+  0.0300; 

+  0.0300. 

46]        NORMAL  EQUATIONS  FOR  SECULAR  VARIATIONS.         91 


I         I 


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CS      O5 


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r  3! 


5».  fl        .       . 

§  § 


co 


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10 


te    oo 


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rH          CO 


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s 


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NORMAL  EQUATIONS  FOE  SECULAR  VARIATIONS.        [46 


CO 


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CO 

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i  i 

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v  .2 

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46]       NORMAL  EQUATIONS  FOR   SECULAR  VARIATIONS. 


CO 


O  O  O  CO 

O  b-  JO  CO  O  b- 

H  rH  rH  CM  rH  rH 

I  +  I  I  +  I 


1  1  1          + 


b'38 


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94 


SECULAR  VARIATIONS  FROM  OBSERVATIONS. 


[46 


Values  of  the  secular  variations  as  derived  from  observations 

only. 


Unknown.        Corr. 


Tables. 


Result. 


Mercury.       Dt  e 


Dti 


-.0691  -0.83  +     4.19  +     3.36i0.50 

+  .1577  +1.30  +116.94  +118.24±0.40 

+  .0593J  +0.83  +     6.31  +     7.14i0.80 

+  .0815K  +0.70  -  92.59  -  91.89±0.50 

-.0967  -1.55 


Yenus. 


Earth. 


Mars. 


Dte        +.1393  +1.67  -  11.13  -     9.46i0.20 

eDt7T      +.0685  +0.82  -     0.53  +     0.29±0.20 

Dt-t        +.1153  J  -0.65  +     4.52  +     3.87i0.30 

sintDt0  -.0592K  -2.73  -102.67  -105.40+0.12 

DttfZ      -.1919  -3.84 

Dte  +0.21  -     8.76  -     8.55i0.09 

eVt7t  +0.26  +  19.22  +  19.48±0.12 

+0.48  -  47.59  -  47.11  =fc  0.25 


Dtf 

Dte 

eDt;r 

Dti 

siniDt 
~Dt6l 


—.1190       -0.68  +  19.68  +  19.OOiO.27 

+.0536       +0.29  +149.26  +149.55±0.35 

+.1136J    +0.17  -     2.43  -     2.26±0.20 

-.0442N  -0.76  -  71.84  -  72.60±0.20 
-.0946       -0.76 


The  first  column  of  numbers  in  this  table  gives  the  unknown 
quantity  as  found  immediately  from  the  eliminating  equations. 
These  quantities  being  multiplied  by  the  factors  given  in 
§  27,  we  have  the  corrections  to  the  tabular  secular  varia- 
tions, as  given  in  the  column  "correction."  The  next  column 
gives  the  value  of  the  tabular  secular  variations,  which  are 
in  all  cases  those  actually  adopted  by  LEVERRIER.  In  the 
case  of  the  Earth,  as  has  been  pointed  out  by  STURMER  and  by 
INNES,  the  secular  variation  of  the  radius  vector  does  not  cor- 
respond to  that  of  the  longitude.  But  as  that  of  the  longitude 
is  the  preponderating  quantity  in  its  effect  on  geocentric 


46,  47]         CORRECTIONS  TO   THE   SOLAK  ELEMENTS.  95 

places,  I  have  regarded  the  value  of  the  eccentr  city  used  in 
the  tables  of  the  equation  of  the  center  as  the  tabular  one  to 
be  adopted. 

The  numbers  in  the  column  "Unknown,"  which  are  followed 
by  the  letters  J  and  N,  are  the  respective  values  of  [J]L  and 
[aST]i,  which  are  changed  to  <5i  and  sin  id  8  by  the  equations 
of  §  41. 

Finally,  we  have  the  results  given  in  the  last  column  for  the 
actual  secular  variations  of  the  several  elements  as  derived 
from  the  preceding  discussion  of  all  the  observations. 

The  result  is  followed  by  the  probable  mean  error  of  each  of 
the  quantities  as  estimated  from  the  probable  magnitude  of 
the  sources  of  error  to  which  they  are  liable.  As  in  other 
cases,  these  quantities  are  very  largelv  a  matter  of  judgment, 
because  the  probable  errors  as  determined  in  the  usual  way 
from  the  eliminating  equations  would  be  entirely  unreliable. 

Definitive  corrections  to  the  solar  elements  for  1850. 

47.  Leaving  the  above  results  to  be  subsequently  discussed, 
we  go  on  with  the  solution  of  the  equations.  By  a  continuation 
of  the  process  just  described,  we  regard  the  preceding  secular 
variations  as  known  quantities,  and  substitute  them  in  the 
eliminating  equations  for  the  solar  elements  which  are  derived 
from  the  normal  equations  for  each  planet.  By  this  substitu- 
tion, we  reach  three  fresh  sets  of  values  of  the  corrections  of 
the  solar  elements  themselves,  one  set  from  the  observe tions 
of  each  planet,  which  are  to  be  reduced  to  1850  and  combined 
with  those  already  found  from  observations  of  the  Sun,  in 
order  to  obtain  the  most  probable  result. 

Here  we  meet  with  the  same  difficulty  that  confronted  us  in 
the  case  of  the  secular  variations.  With  the  exception  of  the 
Sun's  mean  longitude,  we  are  to  regard  the  results  derived 
from  each  of  the  planets  as  affected  by  obscure  sources  of 
systematic  error,  the  probable  magnitude  of  which  can  only 
be  inferred  from  the  general  deviation  of  the  quantities  them- 
selves. As  in  the  former  case,  a  is  not  regarded  as  a  quantity 
independently  determined,  but  a-\-  61"  has  been  taken  instead. 
The  concluded  value  of  a  is  "then  found  by  subtracting  61' '/ 
from  61"  -f-  a.  Since  the  corrections  to  the  solar  elements 
pertain  to  each  separate  epoch,  those  derived  from  the  obser- 


96 


ELEMENTS   OF   EARTH'S   ORBIT. 


[47 


vations  of  the  planets  are  severally  reduced  to  1850,  and  the 
results  are  shown  in  the  following  table : 

Separate  values  of  the  corrections  to  the  solar  elements  for  1850, 
after  the  above  definitive  values  of  the  secular  variations  are 
substituted  in  the  eliminating  equations  from  solution  B, 
reduced  to  1850. 


•Je 

*>> 

6f» 

e"6K" 

a  +  <J/" 

a 

From  observations  of  — 
The  Sun 

// 
—  .  30 

// 

+-°5 

(f 

+  .10 

// 
.  oo 

// 
—  .  02 

// 

—  .07 

Mercury 

+•  I3 

H--°7 

+.48 

—  •  47 

-f  .60 

+  -53 

Venus 

-f.  13 

—  .17 

+.06 

—  •  °7 

-f  .34 

+  -5° 

Mars 

+•  25 

+•24 

—.83 

—.82 

+  i.  18 

+  •  94 

Adopted 

—  .  20 

—.02 

-f  .  12 

—  .04 

+  -46 

+  -48 

These  adopted  values  are  employed  in  the  subsequent  stages 
of  the  discussion,  but  are  not  in  all  cases  regarded  as  definitive. 
In  the  case  of  f  the  value  —  0".20  is  that  which  I  have  actually 
used  in  the  subsequent  determinations  of  the  elements,  but  for 
the  final  value  of  the  obliquity  it  will  be  seen  that  I  have 
taken  —  O'MS  as  more  probable. 


CHAPTER  Y. 

MASSES  OF  THE  PLANETS  DERIVED  BY  METHODS  INDE- 
PENDENT OF  THE  SECULAR  VARIATIONS  WITH  THE 
RESULTING  COMPUTED  SECULAR  VARIATIONS. 

48.  The  plan  of  discussion  laid  down  in  Chapter  I  contem- 
plates the  determination  of  the  masses  of  each  of  the  planets 
from  all  data  independent  of  the  secular  variations,  in  order 
to  determine  how  far  the  observed  secular  variations  can  be 
reconciled  with  these  masses.    The  following  is  a  summary  of 
these  determinations.     The  planets  outside  of  Jupiter  need  no> 
discussion,  as  the  well-known  determinations  of  their  masses 
are  amply  accurate  for  all  our  present  purposes. 

Mass  of  Jupiter. 

49.  One  of  the  works  connected  with  the  present  subject  has 
been  the  determination  of  the  mass  of  Jupiter  from  the  motions 
of  (33),  Polyhymnia.    My  work  on  this  subject  has  not  yet  been 
printed  in  full,  but  I  have  given  in  Astronomische  Nachrichteny 
No.  3249  (Bd.  136,  S.  130),  a  brief  summary  of  the  results.    The 
mass  of  Jupiter  has  been  derived  not  only  from  the  motions 
of  Polyhymnia,  but  from  such  other  sources  as  seemed  best 
adapted  to  give  a  reliable  -result.    The  following  table,  tran- 
scribed from  the  publication  in  question,  shows  the  separate 
results  and  the  conclusions  finally  reached : 

Reciprocal  of  mass  of  Jupiter  from —  wt 

All  observations  of  the  satellites,  1047.82  1 

Action  on  FATE'S  comet  (MOL.LER),  1047.79  1 

Action  on  Themis  (KRUEGKER),  1047.54  5 

Action  on  Saturn  (HiLL),  1047.38  7 

Action  on  Polyhymnia,  1047.34  20 

Action  on  WINNECKE'S  comet  (v.  HAERDTL),  1047.17  10 

1047.35 

in.  e.   ±  0.065 

5690  N  ALM 7  97 


98  MASS  OF  JUPITER.  [49 

It  will  be  seen  that  the  result  from  observations  of  the- satel- 
lites has  been  assigned  a  very  small  weight.  This  course  has 
been  indicated  by  the  circumstances.  Other  conditions  being 
equal,  the  greater  the  mass  of  a  planet  the  less  the  propor- 
tionate precision  with  which  that  mass  can  be  determined  by 
observations  on  the  satellites.  In  any  case,  if  the  measures  of 
the  distances  between  the  satellites  and  the  primary  are' in 
error  by  a  small  fraction,  or,  of  their  whole  amount,  then  the 
error  of  the  mass  will  be  in  error  by  the  fraction  3  a  of  its 
amount.  For  reasons  founded  on  the  construction  and  use  of 
the  heliometer,  I  doubt  whether  the  absolute  measures  made 
with  those  forms  of  that  instrument  which  have  been  used  in 
determining  the  mass  of  Jupiter  can  be  relied  upon  within 
their  three-thousandth  part.  If  so,  the  determination  of  the 
mass  of  the  planet  itself  would  be  doubtful  by  its  thousandth 
part  in  each  separate  case.  The  chance  of  personal  equation 
between  transits  of  the  satellites  and  the  planet  vitiates  in  the 
same  way  the  results  from  observed  transits  of  the  planet  and 
satellites.  Notwithstanding  the  great  refinement  of  the  dis- 
cussion by  KEMPF  of  observations  made  at  Potsdam,  and  the 
care  with  which  he,  SCHUR,  and  others  have  determined  the 
mass  of  Jupiter  by  a  discussion  of  all  the^observations  of  the 
satellites,  I  can  not  conceive  that  the  probable  error  of  any 
possible  result  they  could  derive  would  be  less  than  0.3  or  0.4 
in  the  denominator. 

In  this  connection  the  discordances  between"  the  mass  of 
Saturn,  found  by  Prof.  HALL  and  by  other  observers  from 
observations  of  the  satellites,  are  worthy  of  consideration. 
They  lead  us  to  suspect  that  perhaps  it  is  through  good  for- 
tune rather  than  by  virtue  of  their  absolute  reliability  that 
determinations  of  the  mass  of  Jupiter  from  observations  of  the 
satellites  have  agreed  so  well. 

As  to  the  weights  assigned  to  the  other  results,  only  the  last 
needs  especial  mention.  The  probable  error  assigned  by  v. 
HAERDTL  to  his  result  is  very  much  smaller  than  that  which 
I  find  for  the  mean  of  all  the  results.  But,  as  remarked  in  the 
paper  in  question,  it  has  received  a  smaller  relative  weight 
than  that  corresponding  to  its  assigned  probable  error,  because 
of  distrust  on  my  part  whether  observations  on  a  comet  can 


49,  50,  51]  MASSES  OF  THE  EARTH,  MARS,  AND  JUPITER.  99 

be  considered  as  having  always  been  made  011  the  center  of 
gravity  of  a  well-defined  mass,  moving  as  if  that  center  were 
a  material  point  subject  to  the  gravitation  of  the  Sun  and 
planets.  This  distrust  seems  to  be  amply  justified  by  our 
general  experience  of  the  failure  of  comets  to  move  in  exact 
accordance  with  their  ephemerides. 

I  propose  to  accept  the  value  thus  found, 

Mass  of  Jupiter  =  1  4-  1047.35 

as  the  definitive  one  to  be  used  in  the  planetary  theories. 
Mass  of  Mars. 

50.  In  consequence  of  the  minuteness  of  the  mass  of  Mars, 
measures  of  its  satellites,  especially  the  outer  one,  afford  a 
value  of  its  mass  much  better  than  can  be  derived  by  its  action 
on  the  planets.     When  nearest  the  earth,  the  major  axis  of  the 
orbit  of  the  outer  satellite  subtends  an  angle  of  70".     I  can 
not  think  that  the  systematic  error  to  be  feared  in  the  best 
measures,  such  as  those  made  by  Prof.  HALL,  can  be  as  great 
as  half  a  second.     It  therefore  Appears  to  me  that  the  mean 
error  in  adopting  Prof.  HALL'S  value  of  the  mass  does  not 
exceed  its  fiftieth  part.     This  is  a  degree  of  precision  much 
higher  than  that  of  any  determination  through  the  action  of 
Mars  on  another  planet. 

Prof.  HALL'S  measures  of  1892  show  a  minute  increase  of 
the  mean  distance  given  by  his  work  of  1877.     The  result  is — 

v>"  =  +  0.014 

These  observations,  however,  were  made  when  the  position  of 
the  orbit  of  the  satellite  was  unfavorable  to  an  exact  deter- 
mination of  the  elements  of  motion.  I  have  adhered  to  the 
original  value  in  the  work  of  the  present  chapter. 

Mass  of  the  Earth. 

51.  I  have  already  pointed  out  the  difficulty  in  the  way  of 
determining  the  mass  of  the  Earth  from  its  action  on  the 
other  planets.     On  tbe  other  hand,  the  solar  parallax  has,  in 
recent  years,   been  determined  in   various  ways  with   such 
precision  that  the  mass  of  the  Earth  to  be  used  in  the  plan- 


100  MASS   OF   THE   EARTH.  [51 

etary  theories  can  best  be  derived  from  it.  The  theory  of  the 
relation  between  the  mass  of  the  Earth  and  its  distance  from 
the  Sun,  as  given  by  observations  of  the  seconds  pendulum 
and  the  length  of  the  sidereal  year,  is  one  of  the  best  estab- 
lished results  of  celestial  mechanics.  It  is,  in  effect,  the 
principle  on  which  the  lunar  theory  is  constructed.  In  this 
theory  the  disturbing  action  of  the  Sun  is  necessarily  a  func- 
tion of  the  ratio  of  the  mass  of  the  Sun  to  that  of  the  Earth. 
But  in  the  accepted  theory  this  ratio  is  eliminated  through 
the  ratio  of  the  lunar  month  to  the  sidereal  year.  From  the 
well-established  ratio  between  the  distance  of  the  Moon  and 
the  length  of  the  seconds  pendulum,  the  ratio  of  the  masses 
of  the  Sun  and  Earth  come  out  of  this  theory  with  great 
precision.  It  need  not  be  developed  here;  it  will  suffice  to 
give  the  numerical  result,  which  is  that  between  the  ratio  M 
of  the  mass  of  the  Sun  to  that  of  the  Earth  and  the  mean 
equatorial  horizontal  parallax  of  the  Sun  in  seconds  of  arc 
there  exists  the  relation 

7r3M  =  [8.35493] 

I  have  derived  seven  values  of  the  solar  parallax  by  different 
methods,  of  which  the  following  are  the  preliminary  results : 

wt. 
GILL'S  observations  of  Mars,  1877,  8.780  ±  .020      1 

Contact  observations,  transits  of  Venus.  8.704  i  .018      1 

Aberration  and  velocity  of  light,  8.798  i  .005  16 

Parallactic  equation  of  the  Moon,  8.799  ±  .007      5 

Measures  of  small  planets  on  GILL'S  plan,  8.807  i  .007      8 

LEVERRIER'S  method,  8.818  ±  .030      0.5 

Measures  of  Venus  from  Sun's  center,  8.857  i  .022      1 

Mean  result,  n  =  8".802  i  0".005 

I  have  provisionally  taken  this  mean  as  the  most  probable 
value  of  the  solar  parallax  derived  from  all  sources  except  the 
mass  of  the  Earth.  The  above  relation  then  gives 

M  =  332  040 


51,52]  MASS   OF   VENUS.  101 

Taking  for  the  mass  of  the  Moon  1  4-  81.52,  we  have  for  the 
ratio  of  the  combined  masses  of  the  Earth  and  Moon  to  the 
mass  of  the  Sun 


m"  — 


328  016 


a  result  of  which  the  probable  error  may  be  regarded  as  some- 
thing more  than  a  thousandth  part  of  its  whole  amount. 

Mass  of  Venus. 

52.  The  mass  of  Venus  adopted  in  the  provisional  theory, 
to  which  LEVERRIER'S  tables  were  reduced,  was  .000  002  4885 
=  1  +-  401847,  which  is  that  of  LEVERRIER'S  tables  of  Mer- 
cury. In  the  preceding  discussions  the  following  three  factors 
of  correction  to  this  mass  have  been  found : 

From  observations  of  the  Sun      .     .  —  .0118  =t  .0034 

From  observations  of  Mercury     .     .  —  .0121  =t  .0050 

From  observations  of  Mars      ...  —  .0076  =t  .  ( ? ) 

Mean -  .0119  ±  .0028 

The  mean  error  assigned  to  the  result  from  observations  of 
the  Sun  may  be  regarded  as  real,  because  the  result  is  the 
mean  of  a  great  number  of  completely  independent  determina- 
tions, among  which  no  common  error  is  either  a  priori  prob- 
able or  shown  by  the  discordance  of  the  results.  In  the 
case  of  Mercury,  however,  as  already  remarked,  the  effect  of 
systematic  errors  is  such  that,  although  they  are  almost  com- 
pletely eliminated  from  the  result,  the  mean  error  computed 
in  the  usual  way  would  be  misleading.  The  weight  assigned 
is  therefore  largely  a  matter  of  judgment. 

The  fact  that  it  was  necessary  to  introduce  an  empirical 
correction,  with  a  period  of  about  forty  years,  into  the  mean 
longitude  of  Mars,  vitiates  the  determination  of  the  mass  of 
Venus  from  its  action  on  that  planet,  because  one  of  the  prin- 
cipal terms  in  the  action  of  Venus  on  Mars  has  a  period  which 
does  not  differ  from  forty  years  enough  to  make  the  determi- 
nation of  the  mass  independent  of  this  empirical  correction. 
I  have  therefore  assigned  no  weight  to  the  result.  We  thus 


102  MASS  OF  MERCURY.  [52,53 

have  for  the  mass  of  Venus,  as  derived  from  the  periodic  per- 
turbations of  Mercury  and  the  Earth  produced  by  its  action. 

m'  =  1  -^  406  690  i  1140 
Mass  of  Mercury. 

53.  The  mass  of  Mercury  which  I  have  heretofore  adopted, 
1  -=-  7  500  000,  was  rather  a  result  of  general  estimate  than  of 
exact  computation.  The  fact  is  that  the  determinations  of 
this  mass  have  been  so  discordant,  and  varied  so  much  with 
the  method  of  discussion  adopted,  that  it  is  scarcely  possible 
to  fix  upon  any  definite  number  as  expressive  of  the  mass. 
An  examination  of  LEVERRIER'S  tables  of  Venus  shows  that 
with  the  mass  of  Mercury  there  adopted  (1:3  000  000)  Mercury 
frequently  produces  a  perturbation  of  more  than  one  second 
in  the  heliocentric  longitude  of  Venus.  When  the  latter  is 
near  inferior  conjunction,  the  actual  perturbation  will  be  more 
than  doubled  in  the  geocentric  place,  so  that  the  latter  might 
not  infrequently  be  changed  by  1",  even  if  the  mass  of  Mer- 
cury be  less  than  one-half  LEVERRIER'S  value.  It  was  there- 
fore to  be  expected  that  a  fairly  reliable  value  of  the  mass  of 
Mercury  would  be  obtained  from  the  periodic  perturbations 
of  Venus. 

Eeferriug  to  §  27,  it  will  be  seen  that  the  indeterminate  mass 
of  Mercury  appears  in  the  equations  in  the  form 

1+7;, 
3000000 

From  the  solution  B,  §  38,  the  value  of  /*  comes  out 

^  =  _  0.0834 

corresponding  to  a  mass  of  Mercury  of  1 :  7  210  000.  But  in 
a  subsequent  solution  of  the  equations,  when  the  secular  vari- 
ations are  determined  from  theory  and  substituted  in  the 
normal  equation  for  /v,  we  find 

,u  =  -  0.0889 
which  gives 

m  =  1  -4-  7  943  000 

The  work  of  the  present  chapter  is  based  on  the  former 
value. 


53]  MASS  OF  MERCURY.  103 

A  consideration  of  the  probable  error  of  this  result  is  impor- 
tant. The  fortuitous  errors  which  mostly  affect  it  are  of  the 
class  which  I  have  termed  semi- systematic.  Under  this  term  I 
include  that  large  class  of  errors  which,  extending  through  or 
injuriously  affecting  a  limited  series  of  observations,  cause  the 
probable  error  of  a  result  to  be  larger  than  that  given  by  the 
solution  of  the  equations,  but  which,  nevertheless,  like  purely 
accidental  ones,  would  be  eliminated  from  the  mean  result  of 
an  infinite  series  of  observations.  To  this  class  belong  the 
errors  arising  from  personal  equation  in  observing  the  limb  of 
Venus,  or,  what  is  the  same  thing,  a  difference  between  the 
practical  semidiameter  corresponding  to  the  observer  and  that 
adopted  in  the  reductions.  We  may  suppose  that,  during  a 
period  of  several  days,  when  Venus  is  not  far  from  inferior 
conjunction,  its  geocentric  position  is  affected  by  a  perturba- 
tion produced  by  Mercury.  Through  the  error  alluded  to,  all 
the  observations  made  by  any  one  observer,  and  in  fact  all 
that  are  made  anywhere,  may  be  affected  by  a  certain  con- 
stant error  in  Right  Ascension.  Near  another  inferior  con- 
junction the  same  state  of  things  may  be  repeated,  with  the 
perturbation  in  the  opposite  direction.  If,  now,  tne  observa- 
tions were  made  by  the  same  observer,  and  under  the  same 
circumstances,  the  personal  error  would  be  eliminated  from 
the  mean  of  these  two  results  so  far  as  the  mass  of  Mercury  is 
concerned.  But  very  frequently  different  observers  will  have 
made  the  observations  under  the  two  circumstances,  and  dif- 
ferent conditions  will  have  prevailed.  Thus,  it  is  only  through 
the  general  law  of  averages  that  we  can  expect  the  effect  of 
these  fortuitous  but  systematic  errors  to  be  completely  elim- 
inated. That  they  would  be  eliminated  in  the  long  run  is 
evident  from  the  fact  that  there  can  be  no  permanent  rela- 
tion between  the  personal  equations  of  the  observers  and  the 
changes  in  the  action  of  Mercury  upon  Venus.  Moreover, 
Venus  has  been  observed  with  a  fair  degree  of  accuracy 
through  more  than  half  a  century,  and  it  seems  reasonable 
to  suppose  that  during  that  time  the  errors  in  question  would 
nearly  disappear. 

It  is  clear  from  these  considerations  that  the  probable 
error  derived  from  the  solution  of  the  equations  would  be 


104 


MASS  OF  MERCURY. 


[53 


entirely  misleading.  But  a  probable  error  which  ought  to  be 
reliable  can  be  obtained  by  a  process  similar  to  that  which  I 
have  adopted  elsewhere  in  this  paper,  namely,  dividing  up  the 
materials  into  periods,  and  determining  the  probable  error  from 
the  discordances  among  the  results  of  the  several  periods. 
This  probable  error  will  be  reliable,  because  there  is  no  reason 
why  the  same  error  should  affect  the  mass  of  Mercury  through 
any  two  periods.  I  therefore  take  the  partial  normal  equa- 
tions in  n  derived  from  Eight  Ascensions  during  the  several 
periods,  substitute  in  them  the  values  of  the  unknown  quanti- 
ties found  from  solution  B,  /*  excepted,  and  thus  form  six- 
teen partial  normal  equations  in  /i.  These  equations  may  be 
changed  into  the  corresponding  equations  of  condition,  of 
weight  unity,  by  dividing  each  by  the  square  root  of  the 
coefficient  of  the  unknown  quantity.  The  residuals  then  left 
when  the  definitive  value  of  the  unknown  quantity  is  substi- 
tuted will  be  those  from  whose  discordance  the  probable  error 
may  be  inferred. 

The  partial  normal  equations  thus  found  from  the  Eight 
Ascensions  are  as  follow: 


1750-'62. 

44; 

i=  -  38 

1830-'40. 

5649; 

i=-  831 

1765-'74. 

1265 

-165 

1840-'49. 

2913 

-   18 

1775-'86. 

15 

-   5 

1849->56. 

2238 

-   49 

1787-'96. 

209 

4-  53 

1857->64. 

4506 

-  129 

1796->06. 

345 

4-  19 

1865-'71. 

7736 

-  265 

1806-'14. 

439 

4-135 

1871-79. 

7062 

761 

1814->19. 

942 

+   2 

1879-'86. 

4958 

-  407 

1820-'30. 

1786 

—  330 

1885-'92. 

9561 

-1306 

Sum:     49668/<=  -4095 

^  =  _  0.0824  i  .019 

The  difference  between  this  value  of  yw,  which  is  obtained 
only  to  find  the  probable  error,  and  that  formerly  found,  arises 
principally  from  the  omission  of  the  declination  equations. 
The  mean  error  corresponding  to  weight  unity  comes  out 

ft  =  ±  4".2 


53]  MASS  OF   MERCURY.  105 

which,  as  anticipated,  is  much  larger  than  that  which  would 
be  given  by  the  discordance  of  the  original  observations. 
This  does  not  mean  that  the  original  observations  are  affected 
by  any  such  mean  error  as  ±  4".2,  but  that  the  discordances 
between  the  16  values  of  /*  are  as  great  as  we  should  expect 
them  to  be  if  the  original  observations  were  absolutely  free 
from  systematic  error,  but  affected  by  purely  accidental  errors 
of  this  mean  amount. 

The  results  of  the  solution  for  the  mass  of  Mercury  may  be 
expressed  in  the  form 

.  1±0.32         d 

~  ' 


7  210  000    '         7  043  000 

In  all  researches  which  have  been  made  on  the  motion  of 
ENCKE'S  comet  by  ENCKE,  VON  ASTEN,  and  BACKLUND,  the 
determination  of  this  mass  has  been  kept  in  view.  The 
results  are,  however,  so  discordant  that,  as  already  remarked, 
scarcely  any  definitive  result  can  be  derived  from  them. 

To  this  statement  there  is,  however,  one  apparent  exeeption. 
In  an  appendix  to  his  very  careful  and  elaborate  discussion  of 
WINNECKE'S  comet,  VON  HAERDTL  has  derived  the  value  of 
the  mass  of  Mercury  from  all  the  return  of  ENCKE'S  comet  as 
worked  up  by  VON  ASTEN  and  BACKLTJND.*  The  only  inter- 
pretation which  I  can  put  upon  his  result  is  this  :  If  we  regard 
the  acceleration  of  the  comet,  which  it  is  supposed  results 
from  all  the  observations  made  upon  it,  as  non-existent,  the 
following  two  masses  of  Mercury  are  derivable  from  the  obser- 
vations : 

1819-1868,  w  =  1  4-  5  648  600  i  2000 
1871-1885,  m  =  1  +-  5  669  700  ±  600  000 

He  also  finds,  from  the  motion  of  WINNECKE'S  comet, 
m  =  1  4-  5  012  842  ±  697  863 


*  Denkschriften  der  Kaiserlichen  Akadeinie   der  Wissenschaften,  Vol. 
56,  p.  172-175.     Vienna,  1889. 


106  MASS   OF  MERCURY.  (53,54 

and  from  four  equations  of  LEVERRIER 

1  4-  5  514  700  ±  100  000 

The  consistency  of  these  results  seems  to  me  entirely  beyond 
what  the  observations  are  capable  of  giving,  and  I  hesitate  to 
ascribe  great  weight  to  them.  Moreover,  the  result  implicitly 
contained  in  these  numbers,  that  the  supposed  secular  accel- 
eration of  the  comet  disappears  when  we  attribute  the  pre- 
ceding mass  to  Mercury,  merits  farther  inquiry. 

The  probable  density  of  the  planet  may  form  a  basis  for  at 
least  a  rude  estimate  of  its  probable  mass.  The  fact  that  the 
Earth,  Yenus,  and  Mars  have  densities  not  very  different  from 
each  other,  while  that  of  the  Moon  is  0.6  the  density  of  the 
Earth,  leads  us  to  suppose  that  Mercury,  being  nearest  to  the 
Moon  in  mass,  has  probably  a  slightly  greater  density.  Its 
diameter  at  distance  unity  has  been  repeatedly  measured  and 
found  to  be  6".6,  or,  roughly  speaking,  three-eighths  that  of  the 
Earth.  Were  its.  density  0.7,  its  mass  would  therefore  be 
about  1  :  9,000,000.  In  view  of  the  fact  that  the  measured 
diameter  is  probably  somewhat  too  small,  these  consider- 
ations lead  us  to  conclude  that  the  mass  is  probably  between 
1:6,000,000  and  1:9,000,000. 

As  the  value  of  the  mass  to  be  used  in  investigating  the 
secular  variations,  I  have  adopted 

v  =      0.08 


1  08 
Mass  of  Mercury  = 


7  500  000 


Secular  variations  resulting  from  theory. 

54.  In  the  Astronomical  Papers,  Vol.  V,  Part  IV,  were  com- 
puted the  secular  variations  of  the  elements  of  the  orbits  in 
question  using,  as  the  basis  of  the  work,  the  values  of  the 


THEORETICAL   SECULAR  VARIATIONS. 


107 


54] 

masses  whose  reciprocals  are  found  in  the  column  A  below. 
In  column  B  are  cited  the  masses  which  I  have  decided  upon. 


A 

B 

Original 

Adpofced 

reciprocal 

reciprocal 

of  mass. 

of  mass. 

V 

Mercury, 

7  500  000 

6  944  444 

+  .080 

Venus, 

410  000 

406  750 

+  .0080 

Earth  +  Moon, 

327  000 

328  000 

-.00305 

Mars, 

3  093  500 

3093500 

0 

Jupiter, 

1047.88 

1047.35 

+  .00050 

Saturn, 

3501.6 

0 

Uranus, 

22756 

0 

Neptune, 

19  540 

0 

In  the  case  of  the  Earth  we  have  to  add  the  secular  varia- 
tion of  the  perihelion  produced  by  the  non-sphericity  of  the 
system  Earth  +  Moon.  For  the  principal  term  I  have  found, 

Dt  e"  d  n"  =  +  0".129 

The  resulting  values  of  the  secular  variations,  expressed  as 
functions  of  v,  v1,  v"j  v1",  are  given  in  the  following  section: 

Theoretical  secular  variations  for  1850. 

Mercury. 

//          //  //  //  //  // 

Dte       =  +     4.22 +0.00i/+  2.8  K'  +  l.lF//-0.1?//"  =  +     4.24 

6Dt7Ti     =+109.36+0.00   +56.8    +18.8     +0.5       =  +  109.76 

Dt?:       =+     6.76  -0.04    -  0.6     -  1.4     +0.0       =+     6.76 

sin  i  Dt ;0  =—  92.12  —0.33    -49.3     -12.2      —1.2       =—92.50 

Venus. 


Dte       =-     9.58  —1.30^+  0.0^'-  4.9*///—0.2v///=-     9.67 

eDtTTt     =+     0.39-0.81+0.0    -3.9     +0.5       =+     0.34 

Dti       =+     3.43+0.76   +  0.0    +  0.0      -0.3       =+     3.49 

sin  iT>t#  =-105.92  +0.26   -29.2    -43.2      -1.2       =-106.00 


108  THEORETICAL  SECULAR  VARIATIONS  [54 

Earth. 

//          //  //  //  // 

T>te       =  -     8.57  -0.12F+  1.3^'  -1.6i'"'  =  -     8.57 

eT>t7t     =  +  19.36-0.18   +  5.8  +1.6       =+  19.39 
Dt*       =-  46.65-0.21   -28.3  -0.7       ==-46.89 

Mars. 

//          //  //  //  //  // 

Dte       =+  18.71  +0.03T/4-  O.lv'4-  2.1v7/  =4-  18.71 

eDtTf!     =  +  148.824-0.06   +  4.6    +21.4  =  +  148.80 

Dti       =-     2.34-0.04+12.0    +0.0     +O.OFx//=-     2.25 

«intDt<y=-  72.43-0.27   -25.1    -  7.4     -1.0       =-  72.63 


CHAPTER  VI. 

EXAMINATION  OF  THE  HYPOTHESES  BY  WHICH  THE 
DEVIATIONS  OF  THE  SECULAR  VARIATIONS  FROM 
THEIR  THEORETICAL  VALUES  MAY  BE  EXPLAINED. 

55.  *The  investigations  of  the  present  chapter  are  founded 
on  a  comparison  of  the  secular  variations  derived  purely  from 
observations  in  Chapter  IV,  with  those  resulting  from  the 
values  of  the  masses  obtained  independently  of  the  secular 
variations  in  the  last  chapter.  For  the  sake  of  clearness, 
these  two  sets  of  secular  variations  and  their  differences  are 
collected  in  the  following  table.  The  mean  errors  assigned  to 
the  theoretical  values  are  those  which  result  from  the  prob- 
able mean  errors  of  the  respective  masses.  They  are  there- 
fore not  to  be  regarded  as  independent.  The  mean  errors 
given  in  the  column  of  differences  are  those  which  result  from 
a  combination  of  those  of  the  other  two  columns.  The  errors 
of  the  observed  quantities  must  not,  however,  be  judged  from 
those  of  the  differences,  because  subsequent  changes  in  the 
masses  of  Mercury,  Venus,  and  the  Earth  may  produce  a 
general  diminution  in  the  discordances. 

Mercury. 

Observation.  Theory.  Diff.  A       \/w. 

//          //  //         //  //        //  // 

Dt<?       +     3.36+0.50  +     4.24  +  .01  -0.88i.50  -0.86    2 
6Dt7r      +  118.24+0.40 +109.76+. 16 +8.4S+. 43     .     .      0 

Dti       +     7.14+0.80  +     6.76+.01  +0.38^.80  +0.38    1J 
siniDt#  -  91.89+0.45  -  92.50+.16  +0.61+.52  +0.23    2.2 

Venus. 

Dte  9.46+0.20-     9.67+.24 +0.21+.31  +0.12  5 

eDt7T      +     0.29 i 0.20  +     0.34+.15  -0.05+.25     .     .  0 

Dt*       +     3.87 ± 0.30  +     3.49i.l4 +0.38i.33 +0.44  3J 

siniDt#  -105.40±0.12  -106.00i.12  +0.60i.l7  +0.52  8 

109 


110  COMPARISON  OF  SECULAR  VARIATIONS.  [55 

JEartli. 

Observation.  Theory.  Diff.  A        Vw. 

//          //  //        //  //        //  // 

Dte       -     8.55 ± 0.09  -      8.57 ±.04  +0.02 ±.10  +0.02    10 
eDt7T      +  19.48±  0.12  +  19.38+ .05 +0.10i. 13     .     . 
Dt£       -  47.lliO.23  -  46.89+.09  — 0.22+.27  -0.46     4£ 

Mars. 

Dte  +   19.OOiO.27  +  18.71+.01  -fO.29i.27  +0.29  3.7 

0Dt7r  4-149.55iO.35  +  148.80+.04  +0.75+.35    ...  0 

Dti  2.26+0.20-      2.25 +.04 -0.01 +  .20 +0.08  5 

siniDt#  -  72.60  +  0.20  -  *2.63+.09  +0.03+.22  -0.17  5 

If  we  multiply  the  mean  errors  given  by  0.6745,  to  reduce 
them  to  probable  errors,  we  shall  see  that  only  four  of  the 
fifteen  differences  are  less  than  their  probable  errors.  The 
deviations  which  call  for  especial  consideration  are  the  follow- 
ing four : 

1.  The  motion  of  the  perihelion  of  Mercury.     The  discord- 
ance in  the  secular  motion  of  this  element  is  well  known. 

2.  The  motion  of  the  node  of  Venus.    Here  the  discordance 
is  more  than  five  times  its  probable  error. 

3.  The  perihelion  of  Mars.    Here  the  discordance  is  three 
times  its  probable  error. 

4.  The  eccentricity  of  Mercury.    The  discordance  is  more 
than  twice  its  probable  error.    It  is  to  be  remarked,  however, 
that  the  probable  error  of  this  quantity  is  very  largely  a 
matter  of  judgment,  and  that  its  value  may  have  been  under- 
estimated. 

The  deviations,  if  not  due  to  erroneous  masses,  may  be 
explained  on  two  hypotheses.  One  is  that  propounded  by 
Prof.  HALL,*  that  the  gravitation  of  the  Sun  is  not  exactly  as 
the  inverse  square,  but  that  the  exponent  of  the  distance  is  a 
fraction  greater  than  2  by  a  certain  minute  constant.  This 
hypothesis  accounts  only  for  the  motions  of  the  perihelia,  and 
not  for  any  other  discordances. 

The  other  hypothesis  is  that  of  the  action  of  unknown 
masses  or  arrangements  of  matter.  Since  the  latter  hypothesis 

*  Astronomical  Journal,  Vol.  XIV,  p.  7. 


55,56]  NON-SPHERICITY   OF   THE   SUN.  Ill 

would  account  for  other  motions  than  those  of  the  perihelia,  it 
might  seem  that  the  existence  -of  the  other  discordances 
tells  very  strongly  in  its  favor.  The  hypotheses  of  possible  dis- 
tributions of  unknown  matter,  therefore,  have  iirst  to  be  con- 
sidered.* 

Hypothesis  of  non-  sphericity  of  the  Sun. 

56.  In  a  case  where  our  ignorance  is  complete,  all  hypotheses 
which  do  not  violate  known  facts  are  admissible.  Beginning 
at  the  center  and  passing  outward,  the  first  question  arises 
whether  the  action  may  not  be  due  to  a  non -spherical  distri- 
bution of  matter  within  the  body  of  the  Sun,  resulting  in  an 
excess  of  its  polar  over  its  equatorial  moment  of  inertia.  The 
theory  of  the  Sun  which  has  in  recent  times  been  most  gener- 
ally accepted  is  that  its  interior  may  be  regarded  as  gaseous, 
or  rather  as  a  form  of  matter  which  combines  the  elasticity 
and  mobility  of  a  gas  with  the  density  of  a  liquid.  Such 
being  the  case,  we  may  conceive  that  vortices  of  which  the 
axes  coincide  with  that  of  rotation  may  exist  in  the  interior 
in  such  a  way  that  the  surfaces  of  equal  density  are  non- 
spherical.  A  very  small  inequality  of  this  sort  would  suffice 
to  account  for  the  motion  of  the  perihelion  of  Mercury. 

This  hypothesis  admits  of  an  easy  test.  Whatever  be  the 
nature  or  amount  of  the  inequality,  a  simple  computation 
shows  that  to  account  for  the  observed  phenomenon  it  is 
necessary  and  sufficient  that  the  equipotential  surfaces  at  the 
surface  of  the  Sun  should  have  an  elliptic! ty  of  rather  more 
than  half  a  second  of  arc.  It  can  not,  I  conceive,  be  doubted 
that  the  visible  photosphere  is  an  equipotential  surface.  We 
have  then  to  inquire  whether  there  is  any  such  ellipticity  of 
the  photosphere  as  that  required  by  the  hypothesis.  This 
question  seems  completely  set  at  rest  by  the  great  mass  of 
heliometer  measures  made  by  the  German  observers  in  con- 
nection with  the  transits  of  Venus  of  1874  and  1882,  which 
have  been  discussed  by  Dr.  AUWERS.  The  general  result  is 

*  After  carrying  out  the  investigations  of  this  chapter,  I  find  that  the 
subject  was  studied  on  similar  lines  by  Dr.  P.  HARZER  nearly  three  years 
ago,  and  that  I  made  certain  suggestions  on  the  subject  to  Dr.  BAUSCH- 
INGER  ten  years  ago.  See  Astrononiacliv  Nachrichten,  Vol.  109,  p.  32,  and 
Vol.  127,  p.  81. 


112  INTRA-MERCURIAL  GROUP.  [56,57 

that  the  mean  of  the  equatorial  measures  are  slightly  less  than 
the  mean  of  the  polar  measures,  the  difference,  however,  being 
within  the  probable  errors  of  the  results.  I  conclude  that 
there  can  be  no  such  n  on-  symmetrical  distribution  of  matter 
in  the  interior  of  the  Sun  as  would  produce  the  observed  effect. 
This  same  conclusion  seems  to  apply  to  matter  immediately 
around  the  photosphere.  An  equatorial  ring  of  planetoids,  or 
gaseous  substances  of  the  required  mass,  very  near  the  photo- 
sphere, would  render  the  equipotential  surfaces  of  the  photo- 
sphere elliptical  to  a  degree  which  seems  precluded  by  the 
measures  in  question.  At  a  very  short  distance  from  the  sur- 
face, however,  the  effect  would  be  inappreciable. 

Hypothesis  of  an  intra-mercurial  ring  or  group  of  planetoids. 

57.  Passing  outward,  we  have  next  to  consider  the  hypothe- 
sis of  an  intra-mercurial  ring  adequate  to  produce  the  observed 
phenomena.  In  a  first  approximation  we  may  suppose  the 
ring  circular.  Its  mass  can  not  be  determined,  because  it  will 
depend  upon  the  distance  ;  we  have  to  determine  a  certain 
function  of  the  mass  and  distance  adequate  to  produce  the 
observed  motion  of  the  perihelion.  Then  we  must  inquire  what 
effect  the  ring  will  have  on  the  secular  variations  of  the  other 
elements,  both  of  Mercury  and  of  the  other  planets,  and  see  if 
these  effects  can  be  reconciled  with  observation.  In  the  com- 
putations I  have  assigned  to  the  excess  of  motion  the  pro- 
visional value  40//.7.  If  the  ring  is  not  very  distant  from  the 
Sun  the  motion  which  it  will  produce  in  the  perihelion  of  a 
planet  whose  mean  motion  is  n  and  whose  mean  distance  is  a 
may  be  represented  in  the  form 


}JL  being  a  function  of  the  mass  of  the  ring  and  of  its  radius, 
which  is  nearly  the  same  for  all  of  the  planets,  so  long  as  the 
radius  of  the  ring  is  only  a  small  fraction  of  the  distance  of 
Mercury.  A  first  approximation  to  /*  is— 

u  =  .  m  r2 


57 )  INTRA-MERCTJRIAL 

m  being  tke  ratio  of  its  mass  to  that  of  the  Sun  alSSTFTEs  radius. 
Multiplying  these  motions  in  the  case  of  the  four  planets  by 
their  eccentricities,  we  find  that  the  hypothetical  ring  will 
produce  the  following  secular  variations : 

Mercury,  Dt  n  —  40.7;  eDt  n  =  8.38 
Venus,  4.6  0.031 

Earth,  1.5  0.025 

Mars,  0.34  0.031 

Owing  to  the  sinallness  of  the  eccentricities  the  effect  is 
insensible,  except  in  the  case  of  Mercury,  so  that  the  ring  will 
not  account  for  the  observed  excess  of  motion  of  the  perihelion 
of  Mars. 

Such  a  ring  will  necessarily  produce  a  motion  of  the  plane 
of  the  orbit  of  Mercury  or  Venus,  or  of  both,  because  it  can 
not  lie  in  the  plane  of  both  orbits. 

Let  us  put  ii  for  its  inclination  to  the  ecliptic,  and  61  for  the 
longitude  of  its  node  on  the  ecliptic;  and  let  us  put,  also, 

Pi  =  ii  sin  #! 
ql  =  i,  cos  6\ 

and  let  j?,  p',  ...  ,  q,  q',  .  .  be  the  corresponding  quan 
tities  for  the  planets.  The  theory  of  the  secular  variations 
then  shows  that  the  ring  will  produce  a  motion  of  the  plane  of 
the  orbit  of  Mercury  given  by  the  equations 

Dt^i  =  *-ll  (9i  ~q}  =  40".7  (ql  -  q) 


Expressing  the  motions  of  p  and  q  in  terms  of  the  motions  of  i 
and  0,  which  is  necessary,  owing  to  the  very  different  weights 
of  the  determination  of  the  motion  of  the  planes  of  Mercury 
and  Venus  in  the  direction  of  these  two  coordinates,  we  have 
5690  N,  ALM 8 


114  INTRA-MERCURIAL   GROUP.  [57 

the  following  expressions  for  these  two  motions,  •  which  we 
equate  to  the  observed  excesses  :* 


-  4.96  4-  26.9  £i  4-  28.4_pt  =  +  0.57  ±  0.50 

-  0.27  +    0.8      4-3.0      =  4-  0.63  ±  0.12 
0.00  4-  28.4      -  26.9      =  +  0.50  ±  0.80 
0.00  4-    3.0      -    0.8      =4-  0.45  ±  0.30 
0.00        0.0  1.5      =  -  0.25  ±  0.25 

Multiplying  the  conditional  equations  thus  formed  by  such 
factors  as  will  make  the  mean  error  of  each  equation  nearly 
±  0".5,  we  have  the  following  conditional  equations  for  p{ 
and  #1 : 

27  q,  4-  28^!  =  +  5.53 

3      +  12      =  4.  3.60 
17      _  16      =4-  0.30 

5     -    1      =  4-  0.77 

0      -    3      =  -  0.50 

The  solution  of  these  equations  gives  very  nearly 


^=4-0.12;         0i=4& 

This  great  inclination  seems  in  the  highest  degree  improbable 
if  not  mechanically  impossible,  since  there  would  be  a  tend- 
ency for  the  planes  of  the  orbits  of  a  ring  of  planets  so 
situated  to  scatter  themselves  around  a  plane  somewhere 
between  that- of  the  orbit  of  Mercury  and  that  of  the  invari- 
able plane  of  the  planetary  system,  which  is  nearly  the  same 
as  that  of  the  orbit  of  Jupiter.  Moreover,  the  motion  of  the 
perihelion  of  Mars  is  still  unaccounted  for  and  that  of  the 
node  of  Venus  only  partially  accounted  for,  as  shown  by  the 
large  residual  of  the  second  equation.  In  fact,  the  great  incli- 
nation assigned  to  the  ring- comes  from  the  necessity  of  repre- 
senting as  far  as  possible  the  latter  motion. 

*  It  will  be  noticed  that  iii  forming  these  equations  I  have  neither  used 
the  final  values  of  the  absolute  terms,  nor  taken  account  of  the  fact  that 
the  observed  motions  of  the  planes  are  referred  to  the  ecliptic.  Changes 
thus  produced  in  the  equations  are  too  minute  to  affect  the  conclusion. 


57,  58]  ZODIACAL   LIGHT.  115 

There  would  of  course  be  no  dynamical  impossibility  in  the 
hypothesis  of  a  single  planet  having  as  great  an  inclination  as 
that  required.  But  I  conceive  that  a  planet  of  the  adequate 
mass  could  not  have  remained  so  long  undiscovered.  Whether 
we  regard  the  matter  as  a  planet  or  a  ring,  a  simple  computa- 
tion shows  that  its  mass,  if  at  the  Sun's  surface,  would  be 

about  ri     that  of  the  Sun  itself,  and  one-fourth  of  this  if  at  a 


distance  equal  to  the  Sun's  radius.  We  may  conceive,  if  we 
can  not  compute,  how  much  light  such  a  mass  of  matter  would 
reflect.  Altogether,  it  seems  to  me  that  the  hypothesis  is 
untenable. 


Hypothesis  of  an  extended  mass  of  diffused  matter  like  that  which 
reflects  the  zodiacal  light. 

58.  The  phenomenon  of  the  zodiacal  light  seems  to  show 
that  our  Sun  is  surrounded  by  a  lens  of  diffused  matter  which 
extends  out  to,  or  a  little  beyond,  the  orbit  of  the  Earth,  the 
density  of  which  diminishes  very  rapidly  as  we  recede  from 
the  Sun.  The  question  arises  whether  the  total  mass  of  this, 
matter  may  not  be  sufficient  to  cause  the  observed  motion. 

So  far  as  the  action  of  that  portion  of  matter  which  is  near 
the  Sun  is  concerned,  the  conclusions  just  reached  respecting 
a  ring  surrounding  the  Sun  will  apply  unchanged,  because  we 
may  regard  such  a  mass  as  made  up  of  rings.  Observation 
seems  to  show  that  the  lens  in  question  is  not  much  inclined 
to  the  ecliptic,  and  if  so  it  would  produce  a  motion  of  the 
nodes  of  Venus  and  Mercury  the  opposite  of  that  indicated 
by  the  observations. 

There  is  another  serious  difficulty  in  the  way  of  the  hypoth- 
esis. A  direct  motion  of  the  perihelion  of  a  planet  may  be 
taken  as  indicating  the  fact  that  the  increase  of  its  gravitation 
toward  the  Sun  as  it  passes  from  aphelion  to  perihelion  is 
slightly  greater  than  that  given  by  the  law  of  the  inverse 
square.  This  increase  would  be  produced  by  a  ring  of  matter 
either  wholly  without  or  wholly  within  the  orbit.  But  if  we 
suppose  that  the  orbit  actually  lies  in  the  matter  composing 
such  a  ring,  the  effect  is  the  opposite;  gravitation  toward  the 


11 6  EXTRA-MERCURIAL  GROUP.  [58,  59,  60 

Sun  is  relatively  diminished  as  the  planet  passes  from  aphelion 
to  perihelion,  and  the  motion  of  the  perihelion  would  be  retro- 
grade. 

It  can  not  be  supposed  that  that  part  of  the  zodiacal  light 
more  distant  from  the  Sun  than  the  aphelion  of  Mercury  is 
even  as  dense  as  that  portion  contained  between  the  aphelion 
and  the  perihelion  distances.  The  result  in  question  must 
therefore  be  due  wholly  to  that  part  of  the  matter  which  lies 
near  to  the  Sun,  and  we  thus  have  all  the  difficulties  of  the 
intra-mercurial  ring  theory,  with  one  more  added. 

Hypothesis  of  a  ring  of  planetoids  between,  the  orbits  of  Mercury 

and  Venus. 

59.  It  appears  that  any  ring  or  zone  of  matter  adequate 
to  produce  the  observed  effect  must  lie  between  the  orbits  of 
Mercury  and  Venus.  Its  assignment  to  this  position  requires 
a  more  careful  determination  of  its  possible  eccentricity. 
There  will  be  six  independent  elements  to  be  determined; 
the  mass,  the  mean  distance,  the  eccentricity,  the  perihelion, 
the  inclination,  and  the  node. 

I  find  that  the  observed  excesses  of  motion  of  the  elements 
of  Mercury  and  Venus  will  be  approximately  represented  by 
elements  not  differing  much  from  the  following: 


Total  mass  of  group 37000000 

Mean  distance 0.48 

Eccentricity  of  orbit 0.04 

Longitude  of  perihelion      .     .     ,    .    »  10° 

Longitude  of  node 35° 

Inclination  to  ecliptic* 7°.5 

Probable  diameter  at  distance  unity  if 

agglomerated  into  a  single  planet    .  3".5 

Considerations  on  the  admissibility  of  the  hypothesis — Possible 
mass  of  the  minor  planets. 

60.  Although  the  preceding  hypothesis  is  that  which  best 
represents  the  observations  of  Mercury  and  Venus,  we  can 
not,  in  the  present  condition  of  knowledge,  regard  it  as  more 
than  a  curiosity.  True,  it  is  plausible  at  first  sight.  Since, 


60]  POSSIBLE   ACTION   OF   THE  MINOR   PLANETS.  117 

as  already  remarked,  any  disturbing  body  of  sufficient  mass 
to  cause  the  observed  excess  of  motion  of  the  perihelion  of 
Mercury  would  change  the  position  of  the  planes  of  the  orbits, 
and  since  observations  give  apparent  indications  of  such  a 
change  in  the  plane  of  the  orbit  of  Venus,  it  might  appear 
that  we  have  here  a  very  good  ground  for  the  view  that  all 
the  motions  are  due  to  the  attraction  of  unknown  masses. 
But  the  great  difficulty  is  that  the  excess  of  motion  of  the 
orbital  planes  is  in  the  opposite  direction  from  what  we  should 
expect.  A  group  of  bodies  revolving  near  the  plane  of  the 
ecliptic  would  produce  a  retrograde  motion  of  the  nodes.  But 
the  observed  excess  is  direct.  A  direct  motion  can  be  pro- 
duced only  in  case  the  orbits  are  more  inclined  than  those  of 
the  disturbed  planet.  In  admitting  such  orbits  we  encounter 
difficulties  which,  if  not  absolutely  insurmountable,  yet  tell 
against  the  probability  of  the  hypothesis. 

The  hypothesis  carries  with  it  the  probable  result  that  the 
excess  of  motion  of  the  perihelion  of  Mars  is  produced  by  the 
action  of  the  minor  planets.  I  have  considered  the  question 
of  this  action  in  an  unpublished  investigation.  From  the  prob- 
able albedo  and  magnitude  of  the  minor  planets  and  the  obser- 
vations of  BARNARD  and  others  on  their  diameters,  I  have 
determined  the  probable  mass  of  each  part  of  the  group  having 
a  given  opposition  magnitude.  The  result  is  that  the  number 
of  these  bodies  having  such  a  magnitude  appears  to  progress 
in  a  fairly  uniform  manner  through  several  magnitudes.  The 
ratio  of  progression  may  lie  anywhere  between  the  limits  2 
and  3.  Up  to  the  limit  3  the  total  mass,  if  continued  on  to 
infinity,  could  not  produce  any  appreciable  effect  on  the  motion 
of  Mars.  But  if  we  suppose  a  larger  ratio  than  3  to  prevail, 
then  the  number  of  planets  of  smaller  magnitude  would  be  so 
numerous  us  to  form  a  zone  of  light  across  the  heavens,  as  may 
readily  be  seen  by  considering  that  the  total  amount  of  light 
reflected  from  the  planets  of  each  order  of  magnitude  would 
form  an  increasing  series,  since  the  ratio  between  the  brillian- 
cies of  two  objects  of  unit  difference  in  magnitude  is  only 
about  2.5.  We  may  therefore  suppose  that  the  faint  band  of 
light  which  is  said  to  be  visible  across  the  entire  heavens  as 
a  continuation  of  the  zodiacal  light,  as  well  as  the  "gegen- 


118  HALL'S  HYPOTHESIS.  [60,61 

schein,"  is  due  to  these  minute  bodies,  and  yet  find  their  total 
mass  too  small  to  produce  any  appreciable  effect. 

Whether  we  can  assign  to  the  components  of  such  a  group 
any  magnitude  so  small  that  they  would  be  individually  invis- 
ible, and  a  number  so  small  that  they  would  not  be  seen 
collectively  as  a  band  of  light  brighter  than  the  zodiacal  arch, 
and  yet  having  a  total  mass  so  large  as  to  produce  the  observed 
effects,  is  a  very  important  question  which  can  not  be  decided 
without  exact  photometric  investigations.  It  is,  however,  cer- 
tain that  if  we  could  do  so  we  should  have  to  suppose  a  very 
unlikely  discontinuity  in  the  law  of  progression  between  each 
magnitude  and  the  number  of  bodies  having  that  magnitude. 
It  must  therefore  suffice  for  our  present  object  that  we  regard 
the  hypothesis  of  such  bodies  as  unsatisfactory. 

Hypothesis  that  gravitation  toward  the  sun  is  not  exactly  as  the 
inverse  square  of  the  distance. 

61.  Prof.  HALL'S  hypothesis  seems  to  me  provisionally  not 
inadmissible.  It  is,  that  in  the  expression  for  the  gravitation 
between  two  bodies  of  masses  m  and  m'  at  distance  r 


Force  = 


the  exponent  n  of  r  is  not  exactly  2,  but  2  +  6,  d  being  a  very 
small  fraction.  This  hypothesis  seems  to  me  much  more 
simple  and  unobjectionable  than  those  which  suppose  the 
force  to  be  a  more  or  less  complicated  function  of  the  relative 
velocity  of  the  bodies.  On  this  hypothesis  the  perihelion  of 
each  planet  will  have  a  direct  motion  found  by  multiplying  its 
mean  motion  by  one-half  the  excess  of  the  exponent  of  grav- 
itation. 
Putting 

n  =  2.000  000  1574 

the  excess  of  motion  of  each  perihelion  of  the  four  inner 
planets  would  be  as  follows.  It  will  be  seen  that  the  evidence 
in  the  case  of  Venus  and  the  Earth  is  negative,  owing  to  the 


01]  LAW  OF   GRAVITATION.  119 

very  small  eccentricities  of  their  orbits,  while  the  observed 
motion  in  the  case  of  Mars  is  very  closely  represented. 


Mercury, 

42.34 

8.70 

Venus, 

16.58 

0.11 

Earth, 

10.20 

0.17 

Mars, 

5.42 

0.51 

An  independent  test  of  this  hypothesis  in  the  case  of  other 
bodies  is  very  desirable.  The  only  case  in  which  there  is  any 
hope  of  determining  such  an  excess  is  that  of  the  Moon,  where 
the  excess  would  amount  to  about  140"  per  century.  This  is 
very  nearly  the  hundred- thousandth  part  of  the  total  motion 
of  the  perigee.  The  theoretical  motion  has  not  yet  been  com- 
puted with  quite  this  degree  of  precision.  The  only  determi- 
nation which  aims  at  it  is  that  made  by  HANSEN.*  He  finds 

Theory.  Obser.                  Diff. 

//  //                 // 

Annual  mot.  of  perigee,     146  434.04;  146  435.60;  ^+1.56 

Annual  mot.  of  node,        -69  676.76  —69  679.62;  -2.86 

The  observed  excess  of  motion  agrees  well  with  the  hypoth- 
esis, but  loses  all  sustaining  force  from  the  disagreement  in 
the  case  of  the  node.  The  differences  HANSEN  attributes 
(wrongly,  I  think)  to  the  deviation  of  the  figure  of  the  Moon 
from  mechanical  sphericity. 

Consistency  of  Hall's  hypothesis  with  the  general  results  of  the 
law  of  gravitation. 

62.  The  law  of  the  inverse  square  is  proven  to  a  high  degree 
of  approximation  through  a  wide  range  of  distances.  The  close 
agreement  between  the  observed  parallax  of  the  Moon  and 
that  derived  from  the  force  of  gravitation  on  the  Earth's  sur- 
face shows  that  between  two  distances,  one  the  radius  of  the 
Earth  and  the  other  the  distance  of  the  Moon,  the  deviation 
from  the  law  of  the  square  can  be  only  a  small  fraction  of  the 

*Darlegung,  etc.:  Abhandhungen  der  Math.-Phys.  Classe  der  Kon.  Sdchsi- 
Bchen  Gesellscltaft  der  Wissenschaften,  vi,  p.  348. 


120  HALL'S  HYPOTHESIS.  [62 

thousandth  part,  or,  we  may  say,  a  quantity  of  the  order  of 
magnitude  of  the  five- thousandth  part. 

Coming  down  to  smaller  distances,  we  find  that  the  close 
agreement  between  the  density  of  the  Earth  as  derived  Iroin 
the  attraction  of  small  masses,  at  distances  of  a  fraction  of  a 
meter,  with  the  density  which  we  might  a  priori  suppose  the 
Earth  to  have,  shows  that  within  a  range  of  distance  extend- 
ing from  less  than  one  meter  to  more  than  six  million  meters, 
the  accumulated  deviation  from  the  law  can  scarcely  amount 
to  its  third  part.  The  coincidence  of  the  disturbing  force  of 
the  Sun  upon  the  Moon  with  that  computed  upon  the  theory 
of  gravitation,  extends  the  coincidence  from  the  distance  of 
the  Moon  to  that  of  the  Sun,  while  KEPLER'S  third  law 
extends  it  to  the  outer  planets  of  the  system.  Here,  however, 
the  result  of  observations  so  far  made  is  relatively  less  pre- 
cise. We  may  therefore  say,  with  entire  confidence,  as  a 
result  of  accurate  measurement,  that  the  law  of  the  inverse 
square  holds  true  within  its  five- thousandth  part  from  a  dis- 
tance equal  to  the  Earth's  radius  to  the  distance  of  the  Sun,  a 
range  of  twenty-four  thousand  times ;  that  it  holds  true  within 
a  third  of  its  whole  amount  through  the  range  of  six  million 
times  from  one  meter  to  the  Earth's  radius;  and  within  a 
small  but  not  yet  well-defined  quantity  from  the  distance  of 
the  Sun  to  that  of  Uranus,  in  which  the  multiplication  is 
twentyfold. 

If  HALL'S  hypothesis  contradicted  these  conclusions  it  would 
be  untenable.  But  a  very  simple  computation  will  show  that, 
assuming  the  force  to  vary  as  r-(*  +  8\  d  being  a  minute  con- 
stant sufficient  to  account  for  the  motion  of  the  perihelion  of 
Mercury,  the  effect  would  be  entirely  inappreciable  in  the  ratio 
of  the  gravitation  of  any  two  bodies  at  the  widest  range  of 
distance  to  which  observation  has  yet  extended.  Although 
the  total  action  of  a  material  point  on  a  spherical  surface  sur- 
rounding it  would  converge  to  zero  when  the  radius  became 
infinite,  instead  of  remaining  constant,  as  in  the  case  of  the 
inverse  square,  yet  the  diminution  in  the  action  upon  a  surface 
no  larger  than  would  suffice  to  include  the  visible  universe 
would  be  very  small. 


63]  CORRECTION  OF  MASSES.  121 

Masses  of  the  planets  which  represent  the  secular  variations  of 
other  elements  than  the  perihelia. 

63.  On  HALL'S  hypothesis  the  secular  variations  of  all  the 
elements  other  than  the  perihelia  will  remain  unchanged. 

Our  next  problem  is  to  consider  the  possibility  of  represent- 
ing the  variations  of  the  other  elements  by  admissible  masses 
of  the  known  planets.  In  §  55  I  have  given  a  comparison  of 
the  secular  variations  as  they  result  from  observations,  with 
their  theoretical  expressions  in  terms  of  corrections  to  a  cer- 
tain system  of  masses.  When  the  equations  thus  formed  are 
multiplied  by  the  factors  Vw,  which  make  the  mean  error  of 
each  equation  unity,  we  have  the  following  system  of  equa- 
tions, in  which  we  put  v  =  10  #: 


Ox 

+     QY' 

+     2  v" 

+  Ov"' 

=  -  1.7 

r  =  -  1.8 

0 

-     1 

-     2 

0 

=  +0.5 

+  0.5 

-  7 

-108 

-  27 

-  3 

=  +0.5 

+  1.1 

-65 

0 

-  24 

-  1 

=  +0.6 

+  0.7 

+25 

0 

0 

-  1 

=  +  1.5 

+  1.3 

+  21 

-234 

-346 

-10 

=  4-4.2 

0.0 

-12 

+   13 

0 

-16 

=  +  0.2 

+  0.1 

-  9 

-123 

0 

-  3 

=  -2.0 

-0.7 

+  1 

0 

+     8 

0 

=  +1.1 

+  1.3 

—   2 

+  60 

0 

0 

=  +  0.4 

-0.2 

-14 

-126 

-  37 

-  5 

=  -0.8 

-0.2 

The  resulting  normal  equations  are 

5766  x  —      1563  v1  —      4991  v"  +    140  v1"  =  +    114 

-1563     +101231       +    88556       +3455  •    670 

-  4991     +    88556       +  122462       +  3750         =  -  1446 

+    140     +      3455       +      3750       +    401  39 

Along  with  the  results  of  the  solution  of  these  equations  I 
place,  for  comparison,  the  values  of  Chapter  Y,  which  have 
been  considered  most  probable. 

From  sec.  var.  From  other  sources. 

Wx  =  v     =  +  0.070  +  0.08        ±  0.20 

v'    =  +  0.0100  ±  .0056         +  0.0084    ±  0.0028 

v"  =  -  0.0183  ±  .0052         -  0.00304  i  0.0015 

v"1  =  -  0.0115  ±  .067  +  .0037      i  0.018 


122  CORRECTION  OF  MASSES.  [63, 64 

By  substitution  in  the  conditional  equations  we  find  for  the 
mean  error  corresponding  to  weight  unity— 


*i  =  i  1.14 

In  forming  these  equations  they  were  reduced  by  multipli- 
cation to  a  supposed  mean  error  of  ±  1.  Speaking  in  a 
general  way  we  may  therefore  say  that  the  representation. of 
the  secular  variations,  those  of  the  perihelia  being  ignored, 
by  these  corrections  to  the  masses  is  satisfactory.  Except  for 
the  large  discordance  in  the  motion  of  the  eccentricity  of 
Mercury  the  mean  error  would  have  been  less  than  unity. 

Comparing  the  two  sets  of  values  we  find  that  the  masses 
of  Mercury,  Venus,  and  Mars  agree  well  with  those  derived 
from  other  sources.  Very  different  is  it  with  the  mass  of  the 
Earth.  The  discordance  is  here  more  than  the  hundredth 
part  of  its  whole  amount,  which  involves  a  discordance  of 
more  than  the  three-hundredth  part  in  the  value  of  the  solar 
parallax.  Let  us  now  proceed  in  the  reverse  order,  and  deter- 
mine the  value  of  the  solar  parallax  from  the  mass  of  the  Earth, 
as  derived  from  the  preceding  data. 

Preliminary  adjustment  of  the  two  sets  of  masses. 

64.  We  make  the  best  adjustment  for  this  purpose  by  adding 
to  the  equations  of  condition  last  given  the  additional  ones 
derived  from  the  values  of  the  masses  discussed  in  Chapter  V. 
Multiplying  each  value  of  v  by  the  factor  necessary  to  reduce 
the  mean  error  of  the  second  member  of  the  equation  to  unity, 
we  have  the  following  conditional  equations : 

50  x     =4-  0.4 
360  v'    =  +  2.9 
50  v1"  =       0.0 
30  v">  =  -f  0.42 

Of  the  last  two  equations  it  may  be  remarked  that  the  first  is 
that  given  by  Prof.  HALL'S  original  mass  of  1877,  while  the 
last  is  derived  by  Dr.  HARSHMAN  from  HALL'S  observations 
of  the  outer  satellite  made  during  the  opposition  of  1892. 


64]  CORRECTION  OF  MASSES.  123 

When  we  add  to  the  normal  equations  already  formed  the 
products  of  these  last  equations  by  the  factors  of  the  unknown 
quantities,  the  system  of  normal  equations  is  as  follows : 

8266  #  -     1563  v1  -     4991  r"  +  140  r"'  =+134 

-1563  +230831  +  88556  +3455  =  +374 

-4991  +  88556  +122462  +3750  =  -1446 

+  140  +     3455  +     3750  +3801   •  ==  -26 

The  solution  of  these  equations  gives  the  following  values  of 
the  unknown  quantities : 

x    =  +  0.0071  i  .0120 
v    =  +  0.071    i  .120 
v1  =  +  0.0084  i  .0024 
Vn  =  _  0.0177  i  .0035 
v'"=  +  0.0027  i  .016 

Here  again  we  note  that,  the  Earth  aside,  the  results  for  the 
masses  are  quite  satisfactory.  The  correction  to  Prof.  HALL'S 
original  mass  of  Mars  is  so  minute  and  so  much  less  than  its 
probable  error  that  we  may  consider  this  value  of  the  mass  to 
be  confirmed,  and  may  adopt  it  as  definitive  without  question. 
The  corrections  to  the  masses  of  Mercury  and  Veuus  are  scarcely 
changed.  The  mean  residual  is  reduced  to 

8  =  i  0.91 

which  is  less  than  the  supposed  value. 

We  have,  therefore,  so  far  as  these  results  go,  no  reason  for 
distrusting  the  following  value  of  the  solar  parallax,  which 
results  from  that  of  the  mass  of  the  Earth  thus  derived: 

7i  =  8".759  ±  ".010 

The  critical  examination  and  comparison  of  this  and  other 
values  of  the  parallax  is  the" work  of  the  next  two  chapters. 


CHAPTER  VII. 

VALUES  OF  THE  PRINCIPAL  CONSTANTS  WHICH  DEFINE 
THE  MOTIONS  OF  THE  EARTH. 

The  Precessional  Constant. 

65.  The  accurate  determination  of  the  annual  or  centennial 
motion  of  precession  is  somewhat  difficult,  owing  to  its  depend- 
ence on  several  distinct  elements,  and  to  the  probable  system- 
atic errors  of  the  older  observations  in  Right  Ascension  and 
Declination.  What  is  wanted  is  the  annual  motion  of  the 
equinox,  arising  from  the  combined  motions  of  the  equator 
and  the  ecliptic,  relative  to  directions  absolutely  fixed  in  space. 
As  observations  can  not  be  referred  to  any  line  or  plane  which 
we  know  to  be  absolutely  fixed,  we  are  obliged  to  assume  that  the 
general  mean  direction  of  the  fixed  stars  remains  unchanged, 
or,  in  other  words,  that  the  stellar  system  in  general  has  no 
motion  of  rotation.  This  is  a  safe  assumption  so  far  as  the 
great  mass  of  stars  of  smaller  magnitude  is  concerned.  But  it 
is  not  on  such  stars  that  we  have  the  earliest  accurate  obser- 
vations. Moreover,  observed  Right  Ascensions  of  these 
fainter  stars  relative  to  the  brighter  ones  are  subject  to  possi- 
ble systematic  errors,  arising  from  the  personal  equation  being 
different  for  brighter  and  fainter  stars.  In  the  case  of  the 
stars  observed  by  BRA.DLEY,  there  is  frequently  such  commu- 
nity of  proper  motion  among  neighboring  stars  that  we  can 
noLbe  quite  sure  that  all  rotation  is  eliminated  in  the  general 
mean.  Under  these  circumstances  we  have  only  to  make  the 
best  use  that  we  can  of  existing  material. 

We  must  also  remember  that  observed  Right  Ascensions  are 
not  directly  referred  to  the  equinox,  but  to  the  Sun,  of  which 
the  error  of  absolute  mean  Right  Ascension  must  be  deter- 
mined. This  again  can  be  done  only  from  observed  declina- 
tions, since  by  definition  the  equinox  is  the  point  at  which 
the  Sun  crosses  the  equator.  It  is  also  to  be  noted  that  the 
clock  stars  which  are  directly  compared  with  the  Sun  by  no 
124 


65]  THE  PRECESSIONAL   CONSTANT.  125 

means  include  the  whole  list  to  be  used  as  absolute  points  of 
reference.  We  therefore  have  three  separate  steps  in  determin- 
ing completely  a  correction  to  the  adopted  annual  precession : 

(1)  The  correction  to  tlie  Sun's  absolute  mean  Eight  Ascen- 
sion or  longitude. 

(2)  The  correction  to  the  general  mean  Eight  Ascension  of 
the  clock  stars  relative  to  the  Sun. 

(3)  The  determination  of  the  clock  stars  relative  to  the  great 
mass  of  stars. 

It  goes  without  saying  that  the  determinations  of  these  three 
quantities  are  entirely  independent  of  each  other,  and  that  the 
precision  of  the  result  depends  on  the  precision  of  each  sepa- 
rate determination. 

The  motion  of  the  pole  of  the  equator,  on  which  the  luni- 
solar  precession  depends,  may  be  determined  by  observed 
Declinations  quite  independently  of  the  Eight  Ascensions.  A 
determination  of  the  precession  from  the  latter  includes  the 
planetary  precession,  but  as  this  has  to  be  determined  from 
theory  independently  of  observations,  we  have,  in  observed 
Eight  Ascensions  and  Declinations,  two  independent  methods 
of  determining  the  motion  of  the  equator. 

It  fortunately  happens  that  the  constant  of  precession  is 
not  so  closely  connected  with  other  constants  that  a  small 
error  in  its  determination  will  seriously  affect  our  general  con- 
clusions, or  the  reduction  of  places  of  the  fixed  stars,  because 
the  effect  of  an  error  will  be  nearly  eliminated  through  the 
proper  motions  of  the  fixed  stars,  or  the  motions  of  the  planets 
in  longitude.  I  have  therefore  satisfied  myself  with  reviewing 
and  combining  ,the  four  best  determinations. 

I  pass  over  in  silence  the  classic  determinations  of  BESSEL 
and  OTTO  STRUVE,  because  the  material  on  which  they  depend 
has  been  incorporated  in  more  recent  works.  Of  these  the  one 
which  seems  entitled  to  most  weight  is  that  of  Luowia  STRUVE, 
Bestimmung  der  Constante  der  Prcecession,  und  der  eigenen 
Bewegung  des  Sonnensy 'stems.*  This  work  was  suggested  by 
the  completion  of  AUWERS'  re-reduction  of  BRADLEY'S  Obser- 
vations, and  of  the  Pulkowa  standard  catalogues  for  1845, 

*Me"moires  de  PAcademie  Impe'riale  des  Sciences  de  St.  Pe'tersbourg. 
VIIe  SSrie.  Tome  xxxv,  No.  3. 


126  THE  PRECESSIONAL   CONSTANT.  [65 

1855,  aiid  1865.  It  depends  entirely  on  the  BRADLEY  stars, 
and  the  result,  when  reduced  to  the  most  probable  equinox, 
may  be  regarded  as  the  best  now  derivable  from  those  stars, 
or,  at  least,  as  not  susceptible  of  any  large  correction. 

He,  of  course,  includes  in  his  work  the  determination  of  the 
motion  of  the  solar  system  relative  to  the  mass  of  the  stars. 
In  addition  to  this,  the  possibility  of  a  common  rotation  of 
the  BRADLEY  stars  around  the  axis  of  the  Milky  Way  is  con 
sidered.  This  rotation  I  should  be  disposed  to  regard  as  zero 
for  the  present. 

In  place  of  considering  each  of  the  2,509  stars  singly,  he 
divides  the  celestial  sphere  into  120  spherical  trapezoids,  each 
covering  15  degrees  in  Declination,  and  an  arc  of  Right 
Ascension  equal  approximately  to  one  hour  of  a  great  circle 
at  the  equator.  The  question  might  be  legitimately  raised 
whether  a  different  system  of  weighting  the  trapezoids,  founded 
on  a  consideration  and  comparison  of  the  proper  motions  in 
Eight  Ascension  and  Declination  would  not  have  been  advis- 
able. I  am,  however,  fairly  confident  that  no  change  in  this 
respect  would  have  materially  affected  the  result.  With  this 
work  of  STRUVE  I  have  combined  those  of  BOLTE,  DREYER, 
and  NYREN. 

In  the  case  of  the  Eight  Ascensions  it  is  necessary  to  reduce 
all  the  results  to  the  equinox  determined  in  the  last  chapter. 
From  this  chapter  it  appears  that  the  standard  Eight  Ascen- 
sions with  which  the  reduction  of  the  preceding  investigations 
have  been  made  require  a  correction  to  the  centennial  motion 
of  4-  0".30.  Eeducing  each  determination  to  the  equinox  thus 
defined,  we  have  the  following  results  for  the  general  preces- 
sion in  Eight  Ascension  at  the  epoch  1800 : 

L.  STRUVE,  from  the  comparison  of 
AUWERS-BRADLEY  with  the  modern 
Pulkowa  Eight  Ascensions  .  .  .  m  =  46".050l ;  w  =  4 

DREYER,  from  the  comparison  of 
LALANDE'S  Eight  Ascensions  with 
those  of  SCHIELLERUP 46  .0611;  w  =  2 

NYREN,  by  the  comparison  of  BESSEL'S 
Eight  Ascensions  with  those  of 

S'CH JELLERUP 46    .0456 ;   W  =  I 

Mean  46  .0526 


65]  THE  PRECESSIONAL   CONSTANT.  127 

The  weights  here  assigned  are  of  course  a  matter  of  judgment. 
The  general  agreement  of  the  results  is  as  good  as  we  could 
expect. 
From  observed  declinations  we  have — 

L.  STRUVE,   from  the  comparison  of 

AUWERS  -  BRADLEY    with     modern 

Pulkowa  catalogues w  =  20".0495;  iv  =  2 

BOLTE,  from  the  comparison  of  LA- 

LANDE'S  Declinations  with  those  of 

SCILJELLERUP 20    .0537 ;   w  =  1 

Mean 20  .0509 

We  have  now  to  -combine  these  independent  results.  I  pro- 
pose to  call  Precessional  Constant  that  function  of  the  masses 
of  the  SUE,  Earth,  and  Moon,  and  of  the  elements  of  the  orbits 
of  the  Earth  and  Moon,  which,  being  multiplied  by  half  the 
sine  o£  twice  the  obliquity,  will  give  the  annual  or  centennial 
motion  of  the  pole  on  a  great  -circle,  and  being  multiplied  by 
the  cosine  of  tire  obliquity  will  give  the  lunisolar  precession 
at  any  time.  It  is  true  that  this  quantity  is  not  absolutely 
constant,  since  it  will  change  in  the  course  of  time,  through 
the  diminution  of  the  Earth's  eccentricity.  This  change  is, 
however,  so  slight  that  it  can  become  appreciable  only  after 
several  centuries.  If,  then,  we  put 

p,  the  precessional  constant,  we  have,  for  the  annual  general 
precession  in  Eight  Ascension  and  Declination — 

m  =  p  cos2  e  —  H  sin  L  cosec  s 
n  —  Y  sin  £  cos  £ 

L  being  the  longitude  of  the  instantaneous  axis  of  rotation 
of  the  ecliptic,  and  H  its  annual  or  centennial  motion.  From 
the  definitive  obliquity  and  masses  of  the  planets  adopted 
hereafter,  we  find  the  following  values  of  #,  L,  and  £,  for  1800 
and  1850: 


lOg  H  = 

1800. 
1.67372; 

1850. 
1.67341 

L  = 

173°    2'.31; 

173°  29'.68 

£  = 

23    27.92; 

23    27  .53 

128  THE  PRECESSIONAL   CONSTANT.  [65 

We  thus  find  the  following  values  of  p,  the  unit  of  time 
being  100  solar  years: 

From  Eight  Ascensions,         P  =  5490.12;  w  =  2 
From  Declinations,  p  =  5489.44;  w  =  1 

Mean,  p  =  5489//.89 

As  the  data  used  in  STRUVE'S  Investigation  may  be  con- 
sidered of  a  more  certain  kind  than  those  used  by  the  others, 
we  may  compare  these  results  with  those  which  follow  from 
STRUVE'S  work  alone.  They  are 

From  Eight  Ascensions,  P  =  5489.83 

From  Declinations,  p  =  5489.06 

Giving  double  weight  to  the  results  from  the  Eight  Ascen 
sions,  the  results  may  be  expressed  as  follows : 

From  STRUVE'S  investigation,    P  =  5489.57 
From  the  other  two  works,  p  =  5490.18 

Before  concluding  this  investigation,  I  had  adopted  as  a  pre- 
liminary value 

P  =  5489".78 

As  this  result  does  not  differ  from  the  one  I  consider  most 
probable,  5489".S9,  by  more  than  the  probable  error  of  the 
latter,  and  diverges  from  it  in  the  direction  of  the  best  deter- 
mination, I  have  decided  to  adhere  to  it  as  the  definitive 
value. 

The  centennial  value  of  p  is  subjected  to  a  secular  diminu- 
tion of  0".00364  per  century,  owing  to  the  secular  diminution  of 
the  eccentricity  of  the  Earth's  orbit.  We  therefore  adopt 

p  =  5489.78  —  0.00364  T  for  a  tropical  century. 
p  =  5489.90  -  0.00364  T  for  a  Julian  century. 

In  the  use  of  p  I  at  first  neglected  the  secular  variation, 
but  have-  added  its  effect  to  the  results  developed  in  powers 
of  the  time. 


66]  THE   CONSTANT  OF  NUTATION.  129 

Constant  of  nutation  derived  from  observations. 

66.  The  determination  of  this  constant  from  observations  is 
extremely  satisfactory,  owing  to  the  completeness  with  which 
systematic  errors  may  be  eliminated.  If,  with  a  meridian 
instrument,  regular  observations  are  made  through  a  draconitic 
period,  on  a  uniform  plan,  upon  stars  equally  distributed 
through  the  circle  of  Eight  Ascension,  the  observations  being 
made  daily  through  more  than  12  hours  of  Eight  Ascension, 
all  systematic  errors  in  the  determination  of  the  nadir  point 
and  all  having  a  diurnal  or  annual  period  may  be  completely 
eliminated  from  the  constant  in  question.  These  conditions 
are  so  nearly  fulfilled  in  the  observations  with  the  Greenwich: 
transit  circle,  and,  to  a  less  extent,  in  those  with  the  Wash- 
ington transit  circle,  that  the  results  of  the  Wdrk  with  those 
two  instruments  alone  are  entitled  to  greater  weight  than  has 
hitherto  been  supposed.  I  have,  however,  discussed  quite 
fully  all  previous  determinations  of  which  it  seemed  that  the 
probable  mean  error  would  be  less  than  ±  0".10. 

Eeferring  to  the  volume  on  the  subject  to  be  hereafter  pub- 
lished, the  results  of  the  discussion  are  presented  in  the  fol- 
lowing table.  The  weights  are  assigned  on  the  supposition 
that  weight  unity  should  correspond  to  a  mean  error  of  about 
±  0".07,  or  to  a  probable  error  of  ±  0/7.05,  this  probable  value 
being  not  entirely  a  matter  of  computation  from  the  discord- 
ance of.  the  separate  results,  but,  to  a  certain  extent,  a  matter 
of  judgment. 

It.must  be  understood  that  the  results  below  are  not  always 
those  given  by  the  authors  who  are  quoted,  but  that  their  dis- 
cussion has,,  wherever- possible,  been  subjected  to  a  revision  by 
the  introduction  of  modern  data,  or  by  what  seemed  to  me 
improved  combinations.  Thus,  NYREN'S  equations  have  been 
reconstructed  on  a  system  slightly  different  from  his,  and  have 
been  corrected  for  CHANDLER'S  variation  of  latitude.  PETERS'S 
classical  work  has  also  been  corrected  by  the  introduction  of 
later  data,  and  by  a  re-solution  of  his  equations.  The  Green- 
wich and  Washington  results  have  been  derived  from  the  dis- 
cussion in  Astronomical  Papers,  Vol.  II,  Part  VI. 
5690  N  ALM 9 


130  THE  CONSTANT  OF  NUTATION.  [66 

Values  of  the  constant  of  nutation  derived  from  observations. 

BUSCH,  from  BRADLEY'S  observations  with 

the  zenith  sector 9.232  1 

ROBINSON,  from  Greenwich  mural  circles  .     .  9.22  1 

PETERS,  from  Eight  Ascensions  of  Polaris    .  9.214  4 

LUND AHL,  from  Declinations  of  Polaris     .     .  9.236  1.5 

NYREN,  from  v  Urs.  Maj 9.254  3 

"         "     oDraconis 9.242  2.5 

"          "      i  Draconis 9.240  4 

DE  BALL,  from  WAGNER'S  Eight  Ascensions 

of  Polaris 9.162  3 

DEBALL,   from  WAGNER'S  Declinations  of 

Polaris     . 9.213  3 

DEBALL,  from  WAGNER'S  Eight  Ascensions 

of51Cephei 9.252  3 

DEBALL,   from  WAGNER'S  Declinations  of 

51  Cephei 9.227  3 

DEBALL,  from  WAGNER'S  Eight  Ascensions 

of  6  Urs.  Miu 9.208  3 

DEBALL,  from  WAGNER'S  Declinations  of 

d  Urs.  Min 9.263  3 

Greenwich  North-Polar  Distances  of  South- 
ern Stars,  Series  I 9.116  3 

Greenwich  North-Polar  Distances  of  South- 
ern Stars,  Series  II 9.201  3 

Greenwich  North-Polar  Distances  of  North- 
ern Stars,  Series  I 9.204  4 

Greenwich  North-Polar  Distances  of  North- 
ern Stars,  Series  II 9.223  4 

Washington  Transit  Circle,  southern  stars    .  9.217  6 

"  "  u       northern  stars    .  9.177  3 

Greenwich,  Eight  Ascensions  of  Polaris   .     .  9.153  2 

"  Declinations  of  Polaris  .     .     .     .  9.242  2 

"  Eight  Ascensions  of  51  Cephei    .  9.135  2 

"  Declinations  of  51  Cephei   .     .     .  9.162  2 

"  Eight  Ascensions  of  6  Urs.  Min.  9.147  2 

"  Decimations  of  tfUrs.  Min.     .     .  9.235  2 

"  Eight  Ascensions  of  A  Urs.  Min.  9.161  1 

"  Declinations  of  A  Urs.  Min.  9.339  1 


Mean     . 9.210;  wt.  =  72 


66,67]  PRECESSION  AND  NUTATION.  131 

The  mean  error  corresponding  to  weight  unity  when  derived 
from  the  discordance  of  the  results  is  ±  0".068,  while  the 
estimate  was  i  0".070.  We  may  therefore  put,  as  the  resulj 
of  observation — 

N  =  9".210  ±  0".008 


Relations  betiveen  the  constants  of  precession  and  nutation,  and 
the  quantities  on  which  they  depend. 

67.  The  formula  of  precession  and  nutation  have  been 
developed  by  OPPOLZER  with  very  great  rigor  and  with 
great  numerical  completeness  as  regards  the  elements  of  the 
Moon's  orbit,  in  the  first  volume  of  his  Bahnbestimmung  der 
Kometen  und  Planeten,  second  edition,  Leipzig,  1882.  What 
is  remarkable  about  this  Avork  is  that  it  constantly  takes 
account  of  the  possible  difference  between  the  Earth's  axis 
of  rotation  and  its  axis  of  figure,  a  distinction  which  has 
become  emphasized  by  CHANDLER'S  discovery  since  OPPOL- 
ZER wrote.  His  theory  however  fails  to  take  account  of  the 
change  in  the  period  of  the  Eulerian  nutation  produced  by 
the  mobility  of  the  ocean  and  the  elasticity  of  the  Earth.  But 
this  effect  is  of  no  importance  in  the  present  discussion. 

From  OPPOLZER'S  developments,  I  have  derived  the  follow- 
ing expressions,  in  Avhich  the  numerical  coefficients  may  be 
regarded  as  absolute  constants,  so  accurately  determined  that 
no  question  of  their  errors  need  now  be  considered.  These 
results  haA^e  been  derived  quite  independently  of  the  similar 
ones  by  Mr.  HILL  in  the  Astronomical  Journal,  Yol.  XI,  which 
are  themselves  independent  of  OPPOLZER'S  work.  In  these 
formulae  we  have — 

K,  the  constant  of  lunar  nutation  of  the  obliquity  of  the 
ecliptic,  as  defined  by  the  equation  As  =  ~N  cos  &,  and 
expressed  in  seconds  of  arc; 

P,  so  much  of  the  precession  of  the  equinox  on  the  fixed 
ecliptic  of  the  date,  in  seconds  of  arc  and  in  a  Julian 
year,  as  is  due  to  the  action  of  the  Moon ; 

P7,  so  much  of  the  same  precession  as  is  duejo^j^e^ action 
of  the  Sun. 

^  *     ^l  -  ^- 

Of 


132  .  MASS  OF  THE  MOON.  [67,  68 

We  thus  have, 

luni-solar  precession  =  P  +  P7 

f,     the  obliquity  of  the  ecliptic; 

yu,    the  ratio  of  the  mass  of  the  Moon  to  that  of  the  Earth ; 

A,  the  mean  moment  of  inertia  of  the  Earth  relative  to  axes 

passing  through  its  equator; 
C,   the  same  moment  relative  to  its  polar  axis. 

With  these  definitions  we  have, 

General  value.  Special  value  for  1850. 


X  =  [5.40289J  cos  s 
P  =  [5.975052]  cos  f 
P'  =  [3.72509]  c 


0-A 


C 


=  [5.36542! 


C-A 
0 


1  + 


~  A  =  [5.937585]  _JL_  — 
C  J  1  +  /i      C 


;  "^  =  [3.68762] 


C-^. 
0 


The  special  values  for  1850  are  found  by  putting  for  the 
value  of  the  obliquity  of  the  ecliptic  for  1850, 

f  =  22°  27'  31".  1 


The  mass  of  the  Moon  from  the  observed  constant  of  nutation. 

68.  From  the  two  quantities  given  by  observation,  N  and 
P  4-  P'  =  po,  these  equations  enable  us  to  determine  the  two 


unknown  quantities  yu  and 


C  — A 


As    the    easiest  way  of 


showing  the  uncertainty  of  the  Moon's  mass,  arising  from 
uncertainty  of  the  precession  and  nutation,  I  give  the  value  of 
its  reciprocal  corresponding  to  different  values  of  these  quan- 
tities in  the  following  table : 

Reciprocals  of  the  mass  of  the  Moon  corresponding  to  different 
values  of  the  nutation-constant  and  luni-solar  precession. 


A 

i 

*=•"•» 

50.35 
50.36 
50.37 

81.81 
81.86 
81.  91 

81-53 
81.58 
81.63 

81.  25 
81.30 
81-35 

68,  69J  THE   CONSTANT   OF  ABERRATION.  133 

Taking  for  the  constant  of  nutation  the  value  just  found, 

N  =  9//.210  ±  ".068 
and  for  the  luni-solar  precession, 

lh  =  50" .36  ±  ''.006 

we  have,  for  the  reciprocal  of  the  mass  of 'the  Moon  and  its 
mean  error : 

-  =  81.58  ±  0.20 
p 

The  Constant  of  Aberration. 

69.  In  the  determination  of  astronomical  constants  the  inves- 
tigation of  the  constant  of  aberration  necessarily  takes  a  very 
important  place,  not  only  on  its  own  account  but  on  account  of 
its  intimate  connection  with  the  solar  parallax.  A  general 
determination,  founded  on  all  the  data  available,  was  therefore 
commenced  by  me  as  far  back  as  1890,  before  the  fact  of  the 
variation  of  terrestrial  latitudes  had  been  well  established. 
The  successive  discoveries  of  the  law  of  this  variation  by 
CHANDLER  required  such  alterations  in  the  work  as  it  went 
along  that  much  of  it  is  now  of  too  little  value  for  publication 
in  full.  Happily  the  necessity  for  a  new  discussion  of  the  best 
determinations  at  Pulkowa  has  been  done  away  with  by  the 
papers  of  CHANDLER  himself  in  the  Astronomical  Journal. 

Quite  apart  from  the  disturbing  influence  of  the  revolution 
•of  the  terrestrial  pole  upon  the  determination  of  the  constant 
of  aberration,  this  constant  is  itself  the  one  of  which  the  deter- 
mination is  most  likely  to  be  affected  by  systematic  errors. 
In  this  respect  it  is  at  the  opposite  extreme  from  the  constant 
of  nutation.  From  the  very  nature  of  the  case  it  requires  a 
comparison  of  observations  at  opposite  seasons  of  the  year, 
when  climatic  conditions  are  different.  In  most  cases  the 
determination  must  even  be  made  at  different  times  of  day. 
The  effect  of  aberration  on  a  star,  for  example,  is  generally  at 
one  extreme  when  the  star  culminates  in  the  morning,  and  at 
the  other  extreme  when  it  culminates  in  the  evening.  The 
culminations  at  opposite  seasons  of  the  year  are  necessarily 


134  THE  CONSTANT   OF  ABERRATION.  [69 

associated  with  culminations  at  opposite  times  of  the  day. 
Moreover,  in  observations  to  determine  the  constant  of  aber- 
ration from  Declination,  the  stars  which  give  the  largest  coeffi- 
cients are,  for  the  northern  hemisphere,  tbose  near  18h  of  Eight 
Ascension.  Any  error  peculiar  to  the  times  or  seasons  at 
which  these  stars  are  observed  will  therefore  affect  the  result 
systematically. 

Eight  Ascensions  of  close  polar  stars  also  lead  to  a  value  of 
this  constant.  But  the  same  difficulty  still  exists.  In  this 
case  the  maxima  and  minima  of  aberration  occur  when  the 
star  culminates  at  noon  and  midnight.  Not  only  is  the  aspect 
of  the  star  different  at  the  two  culminations,  but  the  effect  of 
any  diurnal  change  in  the  instrument  will  be  transferred  to  the 
final  result  for  the  aberration. 

The  prismatic  method  of  LOEWY  is  free  from  some  of  these 
objections.  But  its  application  is  extremely  laborious,  and  we 
have,  up  to  the  present  time,  only  two  determinations  by  it, 
one  by  LOEWY  himself,  which  is  only  regarded  as  preliminary, 
and  one  by  COMSTOCK,  in  which  a  large  uncertain  correction 
for  personal  equation  was  applied. 

Under  these  circumstances  the  seeking  of  results  derived  by 
methods  of  the  greatest  possible  diversity  is  yet  more  strongly 
recommended  than  in  the  case  of  the  other  astronomical  con- 
stants. I  have  therefore  used  not  only  the  PULKOWA  deter- 
minations, but  all  those  made  elsewhere  which  it  seemed  worth 
while  to  consider.  Notwithstanding  the  great  amount  of  mate- 
rial added  to  NYREN'S  paper  of  1883,  it  will  be  seen  that  the 
probable  error  of  the  final  result  at  which  I  have  arrived  is 
greater  than  that  which  he  assigns  to  his  result.  This  is  a 
natural  consequence  of  combining  so  many  separate  determi- 
nations. The  advantage  is,  however,  that  the  assigned  prob- 
able error  is  more  likely  to  be  the  real  one.  It  is  not  to  be 
supposed  that  any  of  the  systematic  errors  already  indicated 
would  pertain  to  all  observers  and  to  all  instruments.  The 
final  outcome  should  be  a  result  in  which  the  discordances  of 
the  separate  determinations  show  the  probable  values  of  all 
the  actual  errors,  both  accidental  and  systematic. 

Determinations  founded  on  the  Eight  Ascensions  of  circum- 
polar  stars  are  not  affected  by  the  motion  of  the  terrestrial 


69,  70]  THE   CONSTANT   OF  ABERRATION.  135 

axis,  uor  are  those  founded  on  declinations  of  these  stars,  if 
only  the  declinations  are  observed  equally  at  both  culmina- 
tions. But  determinations  founded  on  declinations  of  stars 
from  upper  culmination  only  are  necessarily  aifected  by  this 
cause.  If  however  the  stars  on  which  the  determination  is 
based  extend  through  the  whole  circle  of  Right  Ascension  the 
effect  of  the  cause  in  question  may  be  wholly  eliminated  by  a 
suitable  treatment  of  the  equations  of  condition.  To  practically 
eliminate  the  injurious  effect  it  is  not  even  necessary  to  deter- 
mine the  exact  law  of  variation.  In  fact,  if  the  stars  observed 
are  equally  scattered  in  Eight  Ascension,  the  effect  of  the  varia- 
tion will  be  partially  eliminated  without  taking  account  of  it. 

CHANDLER  has  shown  that  there  are  two  periodic  terms  in 
the  variation  of  latitude,  one  having  a  period  of  one  year,  the 
other  of  four  hundred  and  twenty-seven  days.  I  may  remark 
that  this  combination  is  in  accord  with  my  theory  developed  in 
the  Monthly  Notices  of  the  Royal  Astronomical  Society  for  March, 
1S92.  It  was  there  shown  that  any  minute  annual  change  of 
the  position  of  the  principal  axis  of  inertia  of  the  Earth — a 
change  which  might  be  produced  by  the  motion  of  water,  ice, 
and  air  on  its  surface — would  appear  as  an  annual  term  in  the 
latitude,  six  times  as  great  as  its  actual  amount, 

Values  of  the  constant  of  aberration  derived  from  observations. 

70.  What  I  have  done  since  this  discovery  by  CHANDLER 
has  been  to  reexamine  the  determinations  of  the  constant  of 
aberration  made  from  time  to  time,  to  make  such  corrections 
in  their  bases  as  seemed  necessary,  and  more  especially  to 
determine  the  correction  to  be  applied  to  each  separate  result 
on  account  of  the  periodic  term  in  the  latitude.  No  attempt 
was  made  to  rework  completely  the  original  material,  except 
in  the  case  of  the  results  of  the  Pulkowa  and  Washington 
observations  with  the  prime  vertical  transit.  In  the  case  of 
the  former,  however,  the  preliminary  results  reached  from  time 
to  time  were  so  accordant  with  those  of  CHANDLER  that  it  is 
a  matter  of  indifference  whether  we  regard  them  as  belonging 
to  his  work  or  to  my  own. 

Owing  to  the  very  different  estimates  placed  by  the  astro- 
nomical world  upon  the  Pulkowa  determinations  and  'those 


136  THE  CONSTANT   OF  ABERRATION.  [70 

made  elsewhere,  I  have  used  the  former  quite  apart  from  the 
others.  The  complete  discussion  of  each  separate  value  is 
too  voluminous  for  the  present  publication,  and  is  therefore 
reserved  for  a  more  extended  future  publication.  At  pres- 
ent it  appears  sufficient  to  judge  the  final  result  by  the  general 
discordance  of  the  material  on  which  it  rests,  rather  than  by 
a  separate  criticism  of  each  particular  case. 

In  the  exhibit  of  results  which  follows  it  is  to  be  remarked 
that  NYREN'S  prime  vertical  observations  do  not  receive  a 
weight  as  great,  relative  to  the  other  Pulkowa  determinations, 
as  would  be  given  by  their  assigned  probable  errors.  The 
reason  of  this  course  is  that  one  can  not  be  entirely  confident 
that  the  results  of  any  one  observer  with  this  instrument  are 
free  from  constant  error  arising  from  differences  of  personal 
equation  in  observing  a  bright  and  a  faint  star.  Many  of  the 
Pulkowa  observations  are  necessarily  made  in  the  morning  or 
evening  twilight.  In  the  case  of  an  evening  observation  the 
star  will  therefore  be  much  fainter  on  account  of  daylight 
when  it  transits  over  the  east  vertical  than  it  will  when  it 
transits  over  the  west  vertical  one  or  two  hours  later.  In  the 
case  of  morning  observations  the  reverse  will  be  true.  It  is 
easy  to  see  that  if,  in  consequence  of  this  difference  of  aspect, 
the  observer  notes  the  passage  of  the  faint  image  too  late,  the 
effect  will  be  to  make  the  constant  of  aberration  too  large. 
The  existence  of  this  form  of  personal  equation,  when  transits 
are  recorded  on  the  chronograph,  is  so  well  known  that,  had 
NYREN'S  observations  been  made  in  this  way,  I  should  not 
have  hesitated  to  ascribe  the  large  values  of  his  aberration 
constant  to  this  cause.  Although  it  has  never  been  shown 
that  any  such  personal  equation  exists  when  observations  are 
made  by  eye  and  ear,  as  KYREN'S  were,  yet  when  we  consider 
that  we  are  dealing  with  quantities  amounting  only  to  one  or 
two  huudredths  of  a  second  of  arc,  and  that  a  personal  equa- 
tion of  this  kind,  undiscoverable  by  ordinary  investigation, 
might  affect  the  result  by  this  minute  amount,  we  can  not  but 
have  at  least  a  suspicion  that  his  values  may  be  slightly  too 
large  from  this  cause. 


70!  THE   CONSTANT   OF  ABERRATION.  137 

Separate  results  for  the  constant  of  aberration. 

A.  Standard  Pulkowa  determinations : 

A  b.      wt. 

Observations    with    Vertical    Circle ;    Polaris,    by       n 
PETERS 20.51     2 

Observations  with  Vertical  Circle ;  7  miscellaneous 

stars,  by  PETERS 20.47     2 

Observations  with  Vertical  Circle  5  1863-1870,  Po- 
laris, by  GYLDEN 20.41     2 

Observations  with  Vertical  Circle;  1871-1875,  Po- 
laris, by  XYREN 20.51     2 

Observations  with  Prime  Vertical;  1842-1844,  by 
STRUVE 20.48     4 

Observations   with   Prime  Vertical;   1879-1880  by 

NYREN.     . 20.52     6 

Observations  with  Prime  Vertical;    1875-1879,  by 
NYREN 20.53     1 

Observations  with  Vertical  Circle;   1863-1873,  by 
GYLDEN  and  NYREN .    20.52     2 

WAGNER  :  Transits  of  three  polar  stars      ....     20.48     5 

From  Eight  Ascensions  of  Polaris;  1842-1844,  by 

LINDHAGEN  and  SCHWEIZER 20.50      2 

Mean  result:  20//.493  ±  0".011 

This  result  may  be  regarded  as  identical  with  that  found  by 

NYREN  in  1882. 

B.  Other  determinations: 

Ab.  e  wt. 

AUWERS,    from    observations    with    the        n 

zenith  sector  at  Kew 20.53     ±  .12        0.5 

AUWERS,  from  WANSTED  observations    .     20.46     ±  .12        0.5 

PETERS,  from  BRADLEY'S  observations 
of  y  Draconis  at  Greenwich  with  zenith 
sector,  1750-1754 20.67  0.5 

BESSEL,  from  Eight  Ascensions  observed 
by  BRADLEY  at  Greenwich  ....  20.71  i.071  0.5 

LINDENAU,  from  Eight  Ascensions  of 
Polaris  observed  at  various  observa- 
tories between  1750  and  1816  .  20.45  ±.05  3 


138  THE   CONSTANT   OF  ABERRATION.  |  70 

Separate  results  for  the  constant  of  aberration — Continued. 
B.  Other  determinations — Continued. 

BRINKLEY,  from  observations  of  thirteen        --/ft.  «•/. 

stars  at  Trinity  College,  Dublin,  with        7/ 

the  8-foot  circle 20.46  ±  .10  1 

PETERS,  from  STRUVE'S  Dorpat  observa- 
tions of  six  pairs  of  circumpolar  stars  .  20.36  ±  .07  2 
RICHARDSON,  from  observations  with  the 

Greenwich  mural  circles 20.50  ±  .06  3 

PETERS,  from  Right  Ascensions  of  Polaris 

at  Dorpat .  20.41  6 

LUNDAHL,  from  Declinations  of  Polaris 

at  Dorpat 20.55  5 

HENDERSON  and  MCLEAR,  from  a1  and 

<*2Centauri  .  .  . 20.52  ±.10  1 

MAIN,  from  observations  with  the  Green- 
wich zenith  tube .  20.20  ±  .10  1 

DOWNING,  from  observations  of  ADra- 

conis  with  reflex  zenith  tube  .  ...  .  20.52  ±  .05  4 
XEWCOMB,  from  observations  of  <*Lyra3 

with   the  'Washington  prime  vertical 

transit,  1862-1867 20.46  ±0.4  6 

NEWCOMB,  from  Right  Ascensions  of 

Polaris  observed  with  the  Washington 

transit  circle,  1866-1867 .  20.55  ±.05  :J 

KUSTNER,  from  observations  of  pairs  of 

stars  by  the  TALCOTT  method  .  .  .  20.46  4 

PRESTON,  from  observations  with  the- 

TALCOTT  method   at  Honolulu,  1891- 

1892 20.43  ±.05  4 

LOEWY,  from  his  prismatic  method  .  .  20.45  ±  .04  5 
COMSTOCK,  using  LOEWY'S  method, 

slightly  modified 20.44  3 

KiisTNER,  from  MARCUSE'S  observations, 

1889-1890 20.49  ±.018  4 

WANACH,  from  Pulkowa  prime  vertical 

observations  .  20.40       ±.015       4 


70,  71]  THE  LUNAR  INEQUALITY.  139 

Separate  results  for  the  constant  of  aberration— Continued. 
B.  Other  determinations — Continued. 

Ab.         tvt. 

From  Greenwich  Eight  Ascensions  of  polar  stars  „ 

with  the  transit  circle 20.39       3 

BECKER,  from  observations  at  Strasburg  by  the 

TALCOTT  method,  1890-1893 20.47       6 

DAVIDSON,    from    similar    observations    at   San 

Francisco,  1892-1894      • 20.48        6 

Mean  result  of  B :  Ab.  const.  =  20".463  ±  0".013 

The  two  results.  A  and  B,  differ  by  0".030,  a  quantity  so 
much  greater  than  their  mean  errors  as  to  leave  room  for  a 
suspicion  of  constant  error  in  one  or  both  means. 

The  Lunar  Inequality  in  the  Ear  til's  motion. 

71.  The  source  of  this  inequality  is  the  revolution  of  the 
center  of  the  Earth  around  the  center  of  mass  of  the  Earth 
and  Moon.  The  former  center  describes  an  orbit  which  is 
similar  to  that  of  the  Moon  around  the  Earth.  Since  this 
orbit  is  not  a  Keplerian  eclipse,  but  is  affected  by  all  the  per- 
turbations of  the  Moon  by  the  Sun,  no  such  element  as  a  semi- 
major  axis  can  be  assigned  to  it.  Instead  of  this  I  take  as  the 
principal  element  of  the  orbit  the  coefficient  of  the  sine  of  the 
Moon's  mean  elongation  from  the  sun  in  the  expression  for  the 
Sun's  true  longitude.  This  element  is  a  function  of  the  solar 
parallax  and  of  the  mass  of  the  Moon,  which  may  be  derived 
from  the  folio wii>g  expression.  Let  us  put 

yw ;  the  ratio  of  the  mass  of  the  Moon  to  that  of  the 

Earth ; 
r,A,  /?;  the  radius  vector,  true  longitude  and  latitude  of 

the  Moon ; 
r',  A',/J';  the  same  coordinates  of  the  Sun; 

* ;  the  linear  distance  of  the  Earth's  center  from  the 
center  of  mass  of  the  Earth  and  Moon. 


140  THE  LUNAR  INEQUALITY.  [71 

We  then  have,  for  the  perturbations  of  the  Sun's  geocentric 
place  due  to  the  cause  in  question  : 

A  log  r1  =  -x  cos  ft  cos  (A-A7) 
A\'  =  -t  cos  ft  sin  (A—  A7) 


and 

s 


I  have  developed  these  expressions,  putting 

7T0  =  8".848 

-.4 

and  taking  for  the  Moon's  coordinates  the  values  found  by 
DELAUNAY.    Putting 

D;  the  mean  value  of  A—  A7 

g,  g1}  the  mean  anomalies  of  the  Moon  and  Sun,  respectively, 
u'-,  the  Sun's  mean  elongation  from  the  Moon's  ascending- 
node; 

the  result  for  JA7  is 

// 

A\'  =      6.533  sin  D 
+  0.013  sin  3  D 
+  0.179  sin  (D  +  g) 

—  0.429  sin  (D  —  g) 
+  0.174  sin  (D  —  g1) 
-  0.064  sin  (D  +  g') 
-f  0.039  sin  (3  D  -  g) 

—  0.014  sin  (D  —  g  —  g1} 

—  0.013  sin  2  u1 

This  value  of  the  lunar  inequality  is  substantially  identical 
with  that  computed  from  the  tables  and  formula  of  LEVER- 


71]  THE   LUNAR   INEQUALITY.  141 

RIER'S  solar  tables.  The  development  of  the  numbers  there 
given  lead  to  the  value  6".534  of  the  principal  coefficient. 

We  have  now  to  find  what  value  of  the  coefficient  is  given 
by  observations.  The  observations  I  make  use  of  are  (1)  all 
the  observations  of  the  Sun's  Eight  Ascension  from  early  in 
the  century  till  1864  ;  (2)  The  heliometer  observations  of  Vic- 
toria made  in  1889  on  GILL'S  plan  and  worked  up  by  him. 

I  had  intended  to  use  all  the  observations  of  the  Sun  up 
to  the  present  time.  1  found  however  that  those  made  after 
1864  gave,  by  comparison  with  the  published  ephemerides, 
inadmissible  positive  corrections  to  the  coefficient.  This  cir- 
cumstance gives  rise  to  a  strong  suspicion  that  in  the  process 
of  interpolating  the  Right  Ascensions  of  the  Sun  during  at 
least  some  years  after  1864,  the  inequality  in  question  was 
rounded  off  to  the  amount  of  several  hundredths  of  a  second. 
The  results  were  therefore  entirely  omitted. 

The  results  for  previous  years,  when  the  inequality  was 
computed  separately  for  every  day  of  observation,  are: 


Greenwich, 

1820-'64; 

-.068 

3.0 

Paris, 

1801-'64; 

-.050 

•   0.8 

Konigsburg, 

1820-'45; 

-.054 

1.2 

Cambridge, 

1828->58; 

-.047 

2.0 

Dorpat, 

1823->38; 

+  .160 

0.3 

Pulkowa, 

1842-'64; 

-.058 

0.5 

Washington, 

1846-'64: 

.000 

0.2 

Mean,  JP  =  —  0".048  i  0".018 

GILL'S  result  is  given  in  the  Monthly  Notices,  Royal  Astro- 
nomical Society,  for  April,  1894  (Vol.  LIY,  page  350.)  It  is 
derived  in  the  following  way.  In  the  solar  ephemeris  which 
he  used  for  comparison  the  lunar  inequalities  were  computed 
rigorously  from  the  coordinates  of  the  Moon,  putting 

n  =  8".880 
M  =  1  -r-  83 

To  the  coefficient  P  thus  arising  he  found  a  correction, 

=  +  0".046 


142  THE   LUNAR   INEQUALITY.  [71,  72 

The  above  values  of  n  and  //  give,  on  the  theory  just  devel- 
oped, 

P  =  6".400 
Thus  GILL'S  result  is,  in  effect, 

P-=  6".446 

while  mine,  from  observations  of  the  Sun,  is 
6".533  —  0".048  =  6".485 

I  consider  that  these  results  are  entitled  to  equal  weight,  and 
that  we  may  take,  as  the  result  of  observation, 

P  =  C>".465  i  0".015 

Solar  parallax  from  the  lunar  inequality. 

72.  With  the  mass  of  the  Moon  already  found  from   the 
observed  constant  of  nutation, 

//  =  1  :  81.58  (1  i  .0025) 

we  may  now  derive  a  value  of  the  solar  parallax  quite  inde 
pendent  of  all  other  values.  The  relation  between  P,  TT,  and 
the  mass  of  the  Moon  is  of  the  general  form 

//  P  =  A'  7T 

where  fc  is  a  numerical  constant,  and,  for  brevity, 


We  have  found  that  the  following  values  correspond  to  one 
theory : 

TT  =  8".848;        X  =  82  5         P  =  6".  533 
Hence  follows 

log  fc  =  1.78207 
so  that  we  have 

^P=  [1.78207]  n 

The  numerical  values  P  =  6/7.465  and  //  =  82.58  now  give 
7r  =  8//.818±  0".030 


73J  PARALLAX  FROM  TRANSITS   OF  VENUS.  143 

Values  of  the  solar  parallax  derived  from  measurements  of  Venus 
on  the  face  of  the  Sun  during  the  transits  of  1874  and  1882, 
with  the  heliometer  and  photoheliograph. 

73.  I  put  these  determinations  into  one  class  because  they 
rest  essentially  on  the  same  principle.  Both  consist,  in  effect, 
in  measures  of  the  distance  between  the  center  of  Yenus  and 
the  center  of  the  Sun  j  the  latter  being  denned  through  the 
visible  limb.  ,  The  method  is  therefore  subject  to  this  serjous 
drawback :  that  the  parallax  depends  upon  the  measured  differ- 
ence between  arcs  which  may  be  from  thirty  to  fifty  times  as 
great  as  the  parallax  itself,  the  measures  being  made  in 
different  parts  of  the  earth. 

The  equations  of  condition  given  by  the  American  photo 
graphs  of  1874  are  found  in  Part  I  of  Observations  of  the 
Transit  of  Yenus,  December  9, 1874 ;  Washington,  Government 
Printing  Office,  1880.  A  preliminary  solution  of  these  equa- 
tions, the  only  one,  however,  to  which  they  have  yet  been  sub- 
jected, was  published  by  D.  P.  TODD,  in  the  American  Journal 
of  Science  for  June,  1881.  (Yol.  XXT,  page  490.) 

The  photographs  of  1882  have  been  completely  worked  up  by 
Professor  HARKNESS,  and  the  results  are  found  in  the  Eeport 
of  the  Superintendent  of  the  Naval  Observatory  for  1889.  The 
equations  derived  from  the  German  heliometer  measures,  with 
a  preliminary  discussion  of  their  results,  are  officially  published 
by  Dr.  AUWERS,  in  the  Bericht  iiber  die  deutschen  Beobachtungen,) 
Y,  p.  710. 

The  separate  results  for  the  parallax,  with  the  probable 
errors  assigned  by  the  investigators,  are  as  follows: 

//                //  w.  W 

1874 :  Photographic  distances,  n  —  8.888  ±  0.040  6  1 

Position  angles,  8.873  ±  0.060  3  3 

Measures  with  heliometer,  8.876  i  0.042  5  5 

1882:  Photographic  distances,  8.847  ±  0.012  64  6 

Position  angles,  8.772  i  0.050  4  4 

Measures  with  heliometer,  8.879  ±  0.025  1C  10 

Under  w  is  given  a  system  of  weights  proportionally  deter- 
mined from  the  probable  errors  as  assigned.  Using  this  sys- 
tem, the  mean  result  is — 

n  =8".854  i  ".016 


144  PARALLAX  FROM   TRANSITS  OF  VENUS.  [73 

I  conceive,  however,  that  these  relative  weights  do  not  cor- 
respond to  the  actual  precision  of  the  measures.  The  very 
small  probable  error  assigned  by  Prof.  HARKNESS  to  the  result 
of  the  photographic  distances  of  1882  does  not  include  the 
probable  error  of  the  angular  value  of  the  unit  of  distance  on 
the  plate,  which  may  arise  from  a  number  of  sources,  includ- 
ing the  possible  deviation  of  the  mirror  of  the  instrument 
from  a  perfect  plane.  From  this  error  the  position  angles 
are  entirely  free.  I  have,  therefore,  assigned  another  set  of 
weights,  w',  which  seem  to  me  to  correspond  more  nearly  to 
the  facts.  The  result  of  this  system  is — 

7t  =  8".857  i  ".016 

This  mean  error  is  derived  from  the  individual  discordances, 
and  not  from  comparisons  with  the  values  of  the  parallax 
otherwise  determined.  As  there  may  be  a  fortuitous  agree- 
ment among  the  separate  values,  another  estimate  may  be 
made  on  the  basis  of  the  total  mean  error  derived  by  AUWERS, 
which  includes  all  known  sources  of  error.  He  finds  6  =  ±  ".032 
for  the  combined  heliometer  results,  to  which  I  have  assigned 
weight  15.  Hence,  for  the  total  weight  29,  we  have — 

<?  =  ±  0".023 

The  deviation  of  the  above  result  from  the  mean  of  all  the 
other  good  ones  is  worthy  of  special  attention.  The  deviation 
is  more  than  three  times  its  mean  error,  and  therefore  between 
four  and  five  times  its  probable  error.  We  must  therefore 
accept  one  of  two  conclusions,  either  the  probable  errors  have 
been  considerably  underestimated,  or  the  method  is  aifected 
with  some  undiscoverable  soured  of  systematic  error,  which 
makes  it  tend  to  give  too  large  a  result.  The  close  accordance 
of  the  six  separate  results,  of  which  only  a  single  one  deviates 
from  the  adopted  mean  by  more  than  its  probable  error,  and 
that  by  only  a  little  more,  would  give  color  to  the  view  that 
the  error  is  a  systematic  one,  and  that  through  some  unknown 
cause  Venus  is  always  measured  too  low  relatively  from  the 
center  of  the  Sun.  I  can  not,  however,  think  of  any  such  cause. 

If  we  determine  the  mean  error  from  the  deviations  of  the 
separate  results  from  what  we  know,  in  other  ways,  to  be 


74)        PARALLAX  FROM  TRANSITS  OF  VENUS.       145 

nearly  the  most  probable  value  of  the  parallax,  namely  8".80, 
we  have — 

Mean  errror  to  weight  1 ;  dz  .148 
Mean  error  of  result  dz.029 

Solar  parallax  from  observed  contacts  during  transits  of  Venus. 

74.  The  contact  observations  of  1761  and  1769  are  discussed 
in  Astronomical  Papers,  Vol.  III.  I  have  also  made  a  com- 
plete discussion  of  those  of  1874  and  1882,  which,  at  the  date 
of  writing,  is  unpublished.  The  separate  results  from  each, 
contact  follow. 

In  the  case  of  the  second  contacts  of  1874  and  1882  it  was 
found  necessary  to  divide  the  observations  into  two  classes: 
those  of  mean  or  true  contact,  and  those  of  the  formation  of 
the  thread  of  light.  In  the  case  of  the  third  contact  no  such 
division  was  necessary,  as  the  observations  could  generally 
be  referred  to  the  same  mean  phase.  The  mean  error  which 
follows  each  result  is  derived  from  the  discordance  of  the 
separate  observations. 

Values  of  the  solar  parallax  from  observed  contacts  of  the  limb 
of  Venus  with  that  of  the  Sun. 

1761,       III;    TT  =  8./78dz/.12;     w.  =    8 
IV;  8.75  dz. 20  3 


1769,          I; 

9.04  d 

z  .17 

4 

II; 

8.55  d 

z  .13 

7 

III; 

8.72  d 

-  .09 

14 

IV; 

9.01  d 

z  .12 

8 

1874,           I; 

8.95  d 

z  .24 

2 

II;  M; 

8.78d 

z  .061 

30 

II;  L; 

8.75  d 

z  .10 

11 

III; 

8.76  d 

z  .045 

57 

IV; 

8.74  d 

-  .09 

14 

1882,          I; 

8.93d 

z  .15 

5 

II;  M; 

8.76d 

z  .042 

64 

II;  L; 

8.72d 

-  .072 

22 

III; 

8.88  d 

-  .042 

64 

IV; 

9.07  d 

-  .12 

8 

5690  N  ALM  10 

146  PARALLAX  FROM  TRANSITS  OF  VENUS.  [74 

The  weights  assigned  are  determined  by  these  mean  errors, 
taken  on  such  a  scale  that  unity  is  the  weight  for  mean  error 
i  ".336.  The  mean  result  of  the  whole  series  is 

7t  =  8".797  i  ".023 

This  mean  error  is  that  resulting  from  the  deviations  of  the 
sixteen  separate  results  from  the  general  mean,  which  give  for 
the  mean  error  corresponding  to  weight  unity, 

£1  =  ±  ".42. 

The  excess  of  this  mean  error  over  that  determined  from  the 
equations  themselves  shows  that  the  general  discordance  of  the 
several  contacts  is  somewhat  greater  than  would  be  inferred 
from  the  individual  discordances  of  the  contacts  inter  se.  This 
is  what  we  should  expect  from  constant  errors  in  the  determi- 
nations of  parallax  from  each  separate  contact.  I  conceive, 
however,  that  such  constant  errors  are  not  likely  to  be  large; 
and  we  can  not  conceive  that  contact  observations  in  general 
are  subject  to  any  constant  error  tending  to  make  the  parallax 
derived  from  them  always  too  great  or  too  small.  I  conclude, 
therefore,  that  the  mean  error  determined  from  the  totality  of 
the  results  may  be  regarded  as  real. 

It  will  be  interesting  to  compare  the  separate  results  of 
internal  and  external  contacts.     They  are 

//  // 

From  internal  contacts ;    TT  ==  8.776  i  .023 
From  external  contacts;    TT  =  8.908  i  .06 

These  mean  errors  are  those  derived  from  the  concluded 
results  and  they  show  that  the  external  contacts  are  relatively 
more  discordant  in  proportion  to  the  weights  assigned  than  are 
the  internal  ones.  If  we  consider  this  discordance  to  indicate 
a  larger  mean  error,  and  therefore  -assign  a  proportionally 
smaller  weight  to  the  results  of  external  contact,  we  have,  for 
the  concluded  result, 

7t  =  8".791  i  ".022 

As  these  two  hypotheses  seem  about  equally  probable,  I  shall 
adopt  the  mean  result, 

n  =  8".794 


75]  PARALLAX  FROM  VELOCITY  OF  LIGHT.  147 

Solar  parallax  from  the  observed  constant  of  aberration  and 
measured  velocity  of  light. 

75.  The  question  of  the  soundness  of  the  proposition  that 
the  aberration  is  equal  to  the  quotient  of  the  velocity  of  the 
Earth  in  its  orbit  by  the  velocity  of  light  is  too  broad  a  one  to 
be  discussed  here.  I  can  only  remark  that  its  simplicity  and 
its  general  accord  with  all  optical  phenomena  are  such  that  it 
seems  to  me  it  should  be  accepted,  in  the  absence  of  evidence 
against  it. 

In  Astronomical  Papers,  Yol.  II,  page  202,  I  have  given  the 
following  determinations  of  the  velocity  of  light  in  vacuo  by 
MICHELSON  and  myself,  expressed  in  kilometers,  per  second : 

4 

MICHELSON  at  Naval  Academy  in  1879 299910 

MICHELSON  at  Cleveland,  1882 299853 

NEWCOMB  at  Washington,  1882,  using  only  results 

supposed  to  be  nearly  free  from  constant  errors     .  299860 

NEWCOMB,  including  all  determinations  .     .    .    ,    .  299810 

I  have  concluded, 

Velocity  of  light  in  vacuo,  =  299860  i  30  k.  m. 

Taking  as  the  equatorial  radius  of  the  Earth  6378.2  k.  m. 
(CLARK),  the  following  table  shows  the  values  of  the  constant 
of  aberration  corresponding  to  admissible  values  of  the  solar 
parallax  when  this  determination  of  the  velocity  of  light  is 
accepted. 

Ab.  =  20.46  n  =  8.8076 

20.47  8.8033 

20.48  8.7990 

20.49  8.7946 

20.50  8.7903 
20.61  8.7859 

20.52  8.7816 

20.53  8.7773 

20.54  8.7730 


148  PAEALLACTIC   INEQUALITY.  [75,  76 

We  thus  have  for  the  values  of  the  solar  parallax  resulting 
from  the  two  values  of  the  constant  of  aberration  already 
derived : 

//  // 

From  Pulkowa  determinations;  Ab.  =  20.493;  n  —  8.793 

From  miscellaneous  determinations;   Ab.  =  20.463;  n  =  8.806 

Solar  parallax  from  the  parallactic  inequality  of  the  Moon. 

76.  I  have  derived  a  value  of  the  parallactic  inequality  of  the 
Moon  from  the  meridian  observations  made  at  Greenwich  and 
Washington  since  1862.  The  determination  of  this  inequality 
is  peculiarly  liable  to  systematic  error,  owing  to  the  fact  that 
observations  have  to  be  made  on  one  limb  of  the  Moon  when 
the  inequality  is  positive,  and  on  the  other  limb  when  it  is 
negative.  Hence,  if  we  determine  the  inequality  by  the  com- 
parison of  its  extreme  observed  effects  on  the  Moon's  longitude 
or  Eight  Ascension,  any  error  in  the  adopted  semidiameter  of 
the  Moon  will  affect  the  result  by  its  full  amount. 

It  does  not  seem  practicable  to  make  a  reliable  determina- 
tion of  the  Moon's  diameter,  because  it  will  necessarily  be 
made  near  the  time  of  full  Moon,  when  the  illumination  of  the 
extreme  limb  is  less  intense  than  near  the  quadratures,  and 
when  some  portions  of  the  limb  that  might  be  visible  if  it  were 
illuminated  by  a  perpendicular  Sun  will  be  thrown  into  shadow 
by  the  horizontal  one.  For  these  reasons  it  may  be  expected 
that  the  parallactic  inequality  determined  by  using  observed 
semidiameters  of  the  Moon  will  be  too  large.  I  have  therefore 
adopted  the  plan  of  determining  the  inequality  from  each  limb 
separately.  To  show  in  regular  progression  the  errors  depend- 
ing on  the  elongation  from  the  Sun,  I  have  classified  the  resid- 
uals of  observations  according  to  the  hour  of  mean  time  at  which 
the  Moon  passed  the  meridian ;  and  formed  equations  of  con- 
dition containing  two  unknown  quantities,  the  one  a  constant 
correction  depending  on  the  semidiameter,  personal  equation, 
etc.,  and  the  other  the  parallactic  inequality.  The  question  is 
further  complicated  by  the  fact  that  the  majority  of  observa- 
tions near  are  quadratures  made  during  daylight,  when  it  is 
to  be  expected  that  the  illumination  of  the  atmosphere  will 


76]  PAKALLACTIC   INEQUALITY.  149 

diminish  the  irradiation,  and  thus  lead  to  a  smaller  apparent 
sernidiameter.  I  have  therefore  sought  to  determine  for  the 
two  observatories,  by  a  comparison  of  the  observations,  the 
correction  to  be  applied  in  order  to  reduce  observations  made 
during  daylight  or  twilight  to  what  they  would  have  been  had 
the  sky  not  been  illuminated.  The  reduction  was  smaller  than 
I  had  expected,  and  somewhat  doubtful  ;  I  have  assigned  pro- 
portionally less  weight  to  those  observations  where  it  was 
necessary.  The  following  are  the  equations  of  condition  thus 
formed.  The  unknown  quantities  are — 

#,  a  constant,   depending  on   the   semidiaineter,   personal 

equation,  etc.; 
2/,  the  correction  to  the  parallactic  inequality  of  the  Moon 

after  reduction  to  the  value  8".848  of  the  solar  parallax. 

GEEE^WICH. 
Limb  I. 


4.6; 
5.6 

x  +  0.93  y   = 
0.99 

-0.53; 
-0.72 

wt.    0.2 
0.6 

6.5 

0.99 

-0.41 

1 

7.5 

0.92 

-0.59 

1 

8.5 

0.79 

-0.54 

1 

9.5 

0.61 

-0.13 

1 

10.5 

0.38 

-0.09 

1 

11.5 

0.13 

-0.06 

1 

Limb  II. 
// 
12.5;  #'-0.13  y  =+0.20;     wt.    1 


13.5 

-0.38 

+  0.16 

1 

14.5 

-0.61 

+  0.28 

1 

15.5 

-0.79 

+  0.54 

1 

16.5 

-0.92 

-  0.11 

1 

17.5 

^0.99 

-0.02 

1 

18.4 

-0.99 

+  0.44 

0.5 

19.4 

—  0.93 

+  1.21 

0.2 

150  PARALLACTIC  INEQUALITY.  [76 

WASHINGTON. 

.    Limb  I. 

h 
4.6 ;  x  +  0.93  y  =  —  1.62  5  ict.  =  0.2 


5.6 

0.99 

-  1.26 

0.4 

6.5 

0.99 

-0.85 

1 

7.5 

0.92 

-0.64 

1 

8.5 

0.79 

-0.71 

1 

9.5 

0.61 

-0.71 

1 

10.5 

0.38 

-  0.48 

1 

11.5 

0.13 

-0.23 

1 

Limb  II. 


12.5; 

x'-  0.13  y 

=  +0.41; 

wt.  =  1 

13.5 

-0,38 

0.43 

1 

14.5 

-0.61 

0.52 

1 

15.5 

-0.79 

0.40 

1 

16.5 

-0.92 

0.72 

1 

17.5 

-0.99 

0.96 

0.5 

18.4 

-  0.99 

1.32 

0.3 

19.4 

-  0.93 

1.50 

0.1 

With  these  equations  we  have  our  choice  to  determine  the 
parallactic  inequality  by  assigning  a  value  to  the  semidiaineter, 
or  to  eliminate  the  semidiameter  from  the  normal  equations. 
In  each  case  the  equations  give  the  following  expressions  for  y: 


Greenwich  :     Limb    I;  y  =  —  0.55  —  1.23  x 
«  "       II;          —0.28+1.230' 

Washington  :  Limb    I;  y  =  —  0.99  —  1.23.r 
"  «       II;  -0.88  +  1.29  a?' 

If  we  choose  to  utilize  the  observed  diameters  we  have  the  fol- 
lowing results: 

From  66  transits  of  the  Moon's  diameter  observed  at  Greenwich; 


76]  PARALLACTIC  INEQUALITY. 

From  33  transits  observed  at  Washington : 


We  should  thus  have, 

From  Greenwich  observations,      y  =  —  0. 
From  Washington  observations,   y  =  —  0.23 

If,  on  the  other  hand,  we  eliminate  x  from  each  pair  of 
normal  equations,  the  final  results  for  y  will  be 

-x/  /x  /f       wf. 

Greenwich :      Limb    I;     0.64  y  =  -  0.45;  y  =  -  0.70  ±  0.16    6 
u      II;     0.64  y  =      0.00 ;  y  =      0.00  i  0.36     2 

Washington :  Limb    I ;     0.64  y  =  -  0.52 ;  y  =  -  0.81  i  0.16     6 
«      II ;     0.53  y  =  -  0.32 ;  y  =  -  0.60  ±  0.27     3 

The  weighted  mean  of  these  results  is 

y=  -  0".64  ±  0".12 
The  resulting  value  of  the  solar  parallax  is 

7t  =  8".802  ±  0".008 

A  very  careful  determination  of  the  solar  parallax  was  made 
from  the  same  theory  by  Dr.  BATTERMAN,  by  means  of  occulta- 
tions,  and  the  result  is  discussed  very  fully  in  the  publica- 
tions of  the  Berlin  Observatory.  Dr.  BATTERMAN'S  definitive 
result  is 

n  =  8".794  ±  ".016 

I  have  slightly  revised  this  result,  by  applying  a  correction 
to  the  coefficient  for  the  parallax  adopted  by  Dr.  BATTERMAN, 
with  the  result 

n  =  8".7S9  i  ".016 

Accepting  this  result,  and  combining  it  with  that  already 
found  from  meridian  observations,  the  parallax  from  this 
method  will  finally  come  out 

n  =  8".799  i  ".007 

This  mean  error  may  be  regarded  as  belonging  to  the  doubtful 
class. 


152  SOLAR  PARALLAX  FROM  MINOR  PLANETS.        [76,77 

While  tin's  work  is  passing  through  the  press  there  appears 
an  important  paper  by  FRANZ  of  Konigsberg,*  giving  the  value 
of  the  parallactic  equation  derived  from  observations  on  the 
lunar  crater  Hosting  A.  The  correction  to  HANSEN'S  coeffi- 
cient is  found  to  be 

-  2".10  i  0".30 

The  corresponding  result  for  the  solar  parallax  is 
8".767  =t  0".021 

We  may  combine  the  three  results  for  the  solar  parallax 
thus  : 


„ 


Greenwich  and  Washington  meridian  obser- 
vations  .     .     ...........  n  =  8.802;  w  =  5 

BATTERMANN  from  occupations  .....         8.789;          2 

FRANZ  from  crater  Hosting  A     .....          8.767;          1 

Mean  ..........          8.794  i  ".008 

Solar  parallax  from  observations  on  minor  planets  with  the 

neliometer. 

77.  The  fact  that  the  determination  of  the  parallaxes  of  the 
small  planets  by  comparison  with  neighboring  stars  is  free 
from  the  grave  uncertainty  attaching  to  similar  observations 
of  Venus  and  Mars,  owing  to  the  absence  of  a  sensible  disk, 
was  long  since  pointed  out  by  Dr.  GALLE.  In  1875  he  pub- 
lished a  discussion  of  observations  on  Flora,  made  at  nine 
northern  observatories,  and  at  the  Cape,  Cordoba,  and  Mel- 
bourne in  the  Southern  hemisphere.t  The  result  was 

n  =  8".873. 

An  examination  of  the  residuals  Of  the  several  observatories 
shows  that  in  the  case  of  at  least  one  of  the  Southern  observa- 
tories there  is  a  systematic  difference  of  a  considerable  fraction 

*  Astronomische  Nachrichten,  Vol.  136,  S.  354. 

tUeber  eine  Bestimnmng  der  Sonnen-Parallaxe  aus  correspondirenden 
Beobachtuugen  des  Planeten  Flora,  in  October  und  November  1873. 
Breslau,  Maruscbke  &  Berendt,  1875. 


77]  SOLAR  PARALLAX  FROM  MINOR  PLANETS.  153 

of  a  second.    This  fact  seems  to  present  our  assigning  any 
appreciable  weight  to  the  final  result. 

In  1874,  GILL,  at  Mauritius,  made  heliometer  observations 
of  Juno,  east  and  west  of  the  meridian,  with  the  same  object. 
The  result  was  8".765,  or  8".815  when  a  discordant  observation 
was  rejected.  In  this  connection,  only  an  allusion  is  necessary 
to  GILL'S  expedition  to  Ascension  in  1877,  made  for  the  pur- 
pose of  applying  the  method  to  Mars  at  the  opposition  of  that 
year. 

Shortly  afterwards  GILL  published  in  the  first  volume  of  The 
Observatory  a  very  exhaustive  discussion  of  the  methods  of 
determining  the  solar  parallax,  in  which  he  showed  that  heli- 
ometer observations  of  the  minor  planets,  made  either  at  a 
single  station  not  too  far  from  the  equator,  or  at  two  stations 
in  different  hemispheres,  afforded  a  method  of  measuring  the 
parallax  more  precise  than  any  before  applied. 

Ten  years  elapsed  before  the  plan  was  put  into  operation. 
Then,  in  1889  and  1890,  a  concerted  system  of  observations  was 
made  on  the  three  minor  planets,  Victoria,  Iris,  and  Sappho,  at 
a  number  of  observatories  in  both  hemispheres.  The  observa- 
tions relating  to  Victoria  were  carried  out  most  thoroughly, 
in  that  a  very  careful  triangulation  of  the  stars  of  comparison 
inter  se  was  made  at  the  observatories  which  took  part  in  the 
measures.  The  tabular  data  for  the  reductions  were  supplied 
by  the  office  of  the  Berliner  Jahrbuch,  and  the  reductions 
and  discussion  were  made  by  GILL  himself  for  Victoria  and 
Sappho,  and  by  Dr.  ELKIN,  on  GILL'S  plans,  for  Iris.  The 
three  results,  as  communicated  in  advance  of  their  complete 
official  publication,  are 

//  // 

From  Victoria:  n  =  8.800  p.  e.  i  0.006 
Iris:  8.825  p.  e.  ±  0.008 

Sappho:  8.796  p.  e.  J=  0.012 

I  assign  the  respective  weights  4,  2,  and  1,  thus  obtaining, 
as  the  final  result  of  this  method, 

n  =  8".807  i  0".006 

I  have  included  in  a  separate  category  GILL'S  determina- 
tion by  Mars,  at  Ascension,  in  1877,  as  published  by  the 


154      UNCERTAINTY  OF  PARALLAX  FROM  MARS.   [77,  78 

Eoyal  Astronomical  Society  (Memoirs  Royal  Astronomical  So- 
ciety, Vol.  XLVI),  for  the  reason  that,  owing  to  the  disk  of 
Mars,  and  its  reddish  color,  determinations  made  on  it  are 
liable  to  errors  peculiar  to  that  planet,  or  at  least  different 
from  those  which  might  come  in  in  the  case  of  the  small 
planets. 

Remarks  on  determinations  of  the  parallax  which  are  not  used 
in  the  present  discussion. 

78.  In  the  preceding  discussion  are  given  the  results  of 
every  modern  method  of  determining  the  solar  parallax  with 
which  I  am  acquainted,  except  meridian  and  eqiiatorial  obser- 
vations on  Mars.  I  have  not  used  any  of  the  results  derived 
from  this  source,  owing  to  their  large  probable  error,  and 
the  suspicion  of  systematic  error  to  which  they  are  open. 
One  of  these  causes  of  error  is  to  be  found  in  the  red  color  of 
Mars.  This  cause  will  be  pointed  out  and  discussed  very 
fully  in  a  subsequent  section.  Its  effect  would  be  to  make  the 
observed  parallax  too  large.  Since,  as  a  matter  of  fact,  all 
the  determinations  of  Mars  by  meridian  observations  have 
given  a  larger  parallax  than  the  generality  of  other  methods, 
color  seems  to  be  given  to  this  suspicion.  Apart  from  this, 
the  setting  of  the  threads  of  a  meridian  circle  upon  the  appar- 
ent disk  of  Mars  involves  a  visual  estimate  not  comparable 
with  that  of  the  bisection  of  the  image  of  a  star  by  the  threads. 
Hence,  there  is  a  chance  of  systematic  personal  error  arising 
from  this  source.  The  observations  generally  exhibit  large 
discordances,  which  may  be  attributed  to  one  or  the  other  of 
these  causes. 

It  may  be  objected  to  the  inclusion  of  GILL'S  Ascension 
result  that  it  should  be  rejected  for  the  same  reason,  since  the 
color  of  the  planet  would  affect  heliometer  observations  and 
meridian  observations  equally.  I  have,  however,  considered 
it  free  from  the  objection  in  question,  for  two  reasons.  In  the 
first  place,  the  result  is  not  too  large,  but  is,  on  the  contrary, 
the  smallest  of  all  the  accurate  measures.  The  principle  that 
when  a  result  is  open  to  a  strong  suspicion  of  being  affected 
by  a  cause  which  would  cause  it  to  deviate  in  one  direction,  it 
is  logical  to  conclude  a  posteriori  that  the  cause  has  not  acted 


78]  UNCERTAINTY  OF   PARALLAX  FROM   MARS.  155 

if  the  deviation  is  found  to  be  in  the  other  direction,  may  not 
be  a  perfectly  sonnd  one,  bnt  I  have  nevertheless  acted  upon 
it.  In  the  next  place  G-ILL  himself,  as  a  part  of  his  discus- 
sion, compared  the  observations  when  Mars  was  at  different 
altitudes,  in  order  to  determine  whether  the  action  of  such  a 
cause  was  indicated,  and  found  a  negative  result. 

In  1890  an  unsuccessful  attempt  was  made,  at  the  writer's 
request,  by  Dr.  W.  L.  ELKIN,  to  measure  the  effect  in  question, 
by  placing  a  refracting  prism  of  very  small  angle  over  one  of 
the  halves  of  a  heliometer  objective,  and  measuring  the  refrac- 
tion thus  produced.  It  was  supposed  that  the  dispersing 
action  of  the  prism  would  represent  that  of  the  atmosphere, 
greatly  magnified.  The  failure  arose  from  the  result  that  the 
apparent  mean  refraction  of  the  star  produced  by  the  prism 
proved  to  be  a  function  of  the  star's  magnitude,  ranging  from 
748".79  for  a  star  of  magnitude  2.55  to  751".61  for  a  star  of  mag- 
nitude 6.95.  The  reason  seemed  to  be  that  too  powerful  a  prism 
was  used,  so  that  the  spectrum  was  quite  sensible;  then,  in  the 
case  of  faint  stars,  the  red  portion  of  the  spectrum  was  invis- 
ible, so  that  the  apparent  mean  refraction  was  greater  than  in 
the  case  of  the  brighter  stars.  The  mean  of  the  observed 
v  displacements  of  Mars  was  748".61,  so  that  it  was  always  less 
for  Mars  than  for  the  stars.* 

An  investigation  of  the  question  whether  the  same  effect  is 
noticeable  in  meridian  observations  fails  to  show  any  relation 
between  the  brightness  of  a  star  and  its  refraction.  But  this 
does  not  disprove  the  relation  between  the  refraction  and  the 
color  of  a  star. 

On  the  whole  it  seems  to  me  that,  at  least  in  the  case  of 
Mars,  we  have  here  a  cause  so  mixed  up  with  personal  error 
in  making  the  observations  that  the  objective  and  subjective 
effects  can  not  be  completely  separated. 

*  Astronomical  Journal,  Vol.  10,  page  97. 


CHAPTER  VIII. 

DISCUSSION    OF    RESULTS    FOR    THE    SOLAR    PARALLAX 
AND  THE  MASSES  OF  THE  FOUR  INNER  PLANETS. 

79.  We  have,  in  what  precedes,  found  or  collected  nine 
separate  values  of  the  parallax  of  the  Sun,  by  methods  of 
which  seven  may  be  regarded  as  completely  distinct,  in  the 
sense  that  no  one  source  of  error  is  common  to  any  two.  Of 
these  seven  the  two  most  nearly  associated  are  those  which 
utilize  transits  of  Venus.  These  are  similar  only  in  the  sense 
of  resting  upon  a  determination  of  the  relative  parallax  of 
Venus  and  the  Sun  during  the  time  of  a  transit.  But  the 
only  common  elements  which  enter  into  the  determination  are 
the  ratio  of  the  distances  of  the  Sun  and  Venus,  which  is 
determined  with  such  certainty  that  we  can  not  regard  it  as 
subject  to  error.  The  methods  of  determining  the  parallax  in 
the  two  cases  are .  completely  distinct  from  the  beginning, 
there  being,  I  conceive,  no  common  source  of  error  affecting 
an  observation  of  contact  of  limbs  and  one  of  a  distance 
measured  from  the  center  of  the  Sun  while  Venus  is  in  transit. 

I  have  classified  as  if  they  were  independent  the  values  of 
the  parallax  which  follow  from  the  Pulkowa  determinations 
of  the  constant  of  aberration,  and  those  which  follow  from  all 
other  determinations.  Of  course  whatever  doubts  may  affect 
the  theory  of  the  assumed  relation  between  the  constant  of 
aberration  and  the  velocity  of  light  will  equally  affect  both 
determinations.  I  do  not,  however,  conceive  that  there  is 
any  source  of  error  which  can  affect  both  the  Pulkowa  deter- 
minations of  the  aberration  and  those  made  elsewhere.  The 
two  could  have  been  combined  so  as  to  give  a  single  result 
of  the  method ;  but  as  the  two  values  of  the  constant  differ 
by  more  than  we  should  expect  them  to  from  their  probable 
errors,  I  have  kept  them  separate,  partly  not  to  give  a  false 
appearance  of  agreement  of  results,  and  partly  to  facilitate 
the  inception  of  any  future  investigation  on  the  subject. 

156 


79]  THE   SOLAR  PARALLAX.  157 

I  have  also  separated  tbe  result  of  GILL'S  observations  on 
Mars,  at  Ascension,  in  1877,  from  the  determinations  made  by 
the  same  method  on  the  minor  planets,  because,  owing  to  the 
color  and  disk  of  Mars,  the  two  results  may  be  affected  by 
very  different  systematic  errors.  The  only  common  systematic 
error  which  seems  likely  to  affect  them  is  that  arising  from  the 
color  of  the  object,  which  will  be  discussed  hereafter. 

Results  of  determinations  of  the  solar  parallax  arranged  in  the 
order  of  magnitude. 

From  the  mass  of  the  Earth  resulting 

from   the   secular   variations  of  the      „  „  wt. 

orbits  of  the  four  inner  planets  .  .  .  8.759  i  .010  9 
From  GILL'S  observations  of  Mars  at 

Ascension  .  .  .  t 8.780  d=  .020  2 

From  Pulkoica  determinations  of  the 

constant  of  aberration 8.793  ±  .0046  40 

From  observations  of  contacts  during 

transits  of  Venus 8.794  i  .018  3 

From  the  parallactic  inequality  of  the 

Moon 8,794  ±  .007  18 

From  determinations  of  the  constant  of 

aberration   made  elsewhere  than  at 

Pulkowa 8.806  ±  .0056  28 

From  heliometer  observations  on  the 

minor  planets 8.807  ±  .007  20 

From  the  lunar  equation  in  the  motion 

of  the  Earth 8.825  i  .030  1 

From  measurements  of  the  distance  of 

Venus  from  t  he  Sun's  center  during 

transits 8.857  i  .023        2 

The  mean  errors  which  follow  each  value  are  those  which, 
from  a  study  of  the  determination,  it  seemed  likely  might 
affect  them,  no  allowance  being  made  for  mere  possibility  of 
systematic  error.  The  weights  assigned  are  convenient  small 
integers,  generally  such  as  to  make  the  weight  unity  corre- 
spond to  the  mean  error  ±  0".30,  allowance  being  made,  how- 


158  THE  SOLAR,  PARALLAX.  [  7j 

ever,  for  doubt  as  to  what  value  should  be  assigned  to  the 
mean  error  and  for  the  different  liabilities  to  systematic  error. 
The  mean  result  is — 

//  // 

From  all  determinations;  n  —  8.797 

Omitting  the  first  result;  n  =  8.800  ±  .0038 

The  last  value  differs  from  the  preliminary  value  8".802  of 
Chapter  V,  from  a  change  in  the  weights.  It  will  be  seen 
that  the  different  values  are  all  as  accordant  as  could  be 
expected,  with  the  exception  of  the  two  extreme  ones.  In  the 
largest  value  we  have  a  case  the  principles  involved  in  which 
have  been  discussed  in  Chapter  IV. 

We  can  not  suppose  the  parallax  to  be  materially  greater 
than  8".800,  and  may  take  it  as  probably  less  than  this.  Thus 
the  absolute  error  of  the  results  of  measures  of  Venus  on  the 
face  of  the  Sun  may  be  considered  as  about  0".06  or  0".07, 
which  is  four  times  the  computed  probable  error.  The  prob- 
ability against  this,  even  in  the  case  of  one  result  out  of  eight 
or  nine,  is  so  small  that  we  must  either  regard  the  method  as 
being  affected  by  some  systematic  error,  or  as  affected  by 
an  objective  probable  error  larger  than  that  assigned.  It 
seems  to  me  the  latter  view  is  not  untenable,  in  view  of  the 
very  wide  range  of  the  possibilities  of  error  which  might  affect 
a  series  of  observations  with  a  heliometer  exposed  to  the  Sun's 
rays  during  a  period  limited  to  a  few  hours. 

Again,  in  the  photographic  measures,  the  value  of  a  second 
of  arc  in  length  on  the  photographic  plate  enters  as  a  some- 
what uncertain  element.  In  this  connection  it  is  to  be 
remarked  that  the  measures  of  position  angle  on  the  photo- 
graphic plates,  which  are  not  affected  with  this  uncertainty, 
although  their  probable  error  is  quite  considerable,  give  a 
value  of  the  solar  parallax  much  smaller  than  the  measures  of 
distance. 

Much  more  embarrassing  is  the  value  which  results  from  the 
mass  of  the  Earth.  We  here  meet  in  another  aspect  the  same 
deviation  which  we  encountered  in  determining  the  mass  of 
the  Earth  from  the  secular  variations,  and  on  which  we  post- 
poned a  conclusion  (§  64).  This  determination  rests  very 


79,  80]  MOTION   OF   THE  NODE   OF  VENUS.  159 

largely  on  the  motion  of  the  node  of  Venus,  as  determined 
from  the  transits  of  1761  and  17G9.  It  is  true  that  results  of 
meridian  observations  are  combined  with  them ;  but  no  expla- 
nation is  thus  afforded  of  the  difficulty,  because  the  results  of 
these  observations  agree  with  those  of  the  transits  (v.  §39). 
What  adds  to  the  embarrassment  and  prevents  us  from  wholly 
discarding  the  suspicion  that  some  disturbing  cause  has  acted 
on  the  motion  of  Venus,  or  that  some  theoretical  error  has 
crept  into  the  work,  is  that,  of  all  the  determinations  of  the 
solar  parallax  this  is  the  one  which  seems  the  most  free  from 
doubt  arising  from,  possible  undiscovered  sources  of  error.  It 
is,  as  we  shall  presently  see,  really  entitled  to  twice  the  relative 
weight  assigned  it.  As,  however,  the  determination  rests 
mainly  on  the  motion  of  the  node  of  Venus,  and  this  again 
mainly  rests  on  the  observations  of  the  older  transits,  I  have 
made  a  reexamination  of  the  results  of  these  transits  with  a 
view  of  reaching  a  more  exact  estimate  of  the  sources  of  error 
and  the  magnitude  of  the  mean  error.  In  this  re-examination 
I  have  regarded  the  Sun's  parallax  as  a  known  quantity  equal 
to  8".798,  and  then  obtained  the  results  of  the  old  observations 
of  the  transits  on  the  supposition  that  the  only  quantities  to 
be  determined  were  the  corrections  to  the  relative  heliocentric 
positions  of  Venus  and  the  Earth. 

Rediscussion  of  the  motion  of  the  node  of  Venus. 

80.  In  discussing  the  observations  of  1761  and  1769  (Astro- 
nomical Papers,  Vol.  II,  Part  V),  I  introduced  a  quantity 
expressive  of  the  error  in  the  observed  time  of  contact  arising 
from  imperfections  of  the  telescope  and  atmospheric  absorp- 
tion and  dispersion.  The  constants  on  which  these  errors 
depend  are  represented  by  symbols  fc2  and  &3.  As  I  have 
worked  up  the  observations,  the  ultimate  result  of  each 
observation  of  contact  is  the  value  of  an  unknown  quantity, 
dCj  which,  were  there  no  imperfections  of  vision  and  were  the 
radii  of  the  Sun  and  Venus  accurately  known,  would  represent 
the  correction  to  the  tabular  distance  of  centers.  As  a  matter 
of  fact,  however,  we  are  to  consider  6  c  as  equal  to  this  correc- 
tion increased  by  a  rather  complex  combination  of  quantities 
depending  on  the  errors  of  the  assumed  semidiameters  of 


160  MOTION  OF  THE  NODE   OF  VENUS.  [80 

Venus  and  the  S.un,  and  the  thickness  of  the  thread  of  light 
when  it  first  became  visible  at  second  contact,  or  vanished  at 
third  contact.  The  observations  must  be  so  combined  as  to 
eliminate  these  quantities.  What  I  have  done  is  to  represent 
the  undiscoverable  minute  correction  to  dc  thus  arising  by 
the  symbol  £2  for  second  contact,  and  03  for  third  contact.  In 
the  present  re-examination  the  absolute  terms  are  reduced  to 
the  parallax  8".798  by  putting  67rQ  =  -  ".05  and  n>  =  -  ".025 
in  the  final  equations  of  the  original  paper.  After  each  result 
is  given  the  mean  .error  with  which  it  is  affected,  as  deter- 
mined by  the  investigation  in  question.  When  thus  treated, 
the  equations  which  I  have  given  on  pages  391-398  of  the 
paper  referred  to  give  the  following  normal  equations  for  tfc, 
the  indeterminates  &2  and  &3  being  retained  as  such  in  order  to 
show  their  final  effect  on  the  result. 

//  // 

1761.    II 5     8.5   do  =  +  0.76  -    18.5  fc2  ±  0.78 
III;  41.7  dc  =  -  2.81  -     19.2  k,  ±  1.30 

1769.     II;  44.8  dc  =  -  8.00  -  104.1  fc2  i  1.95 
III;  12.1  do  =  +  0.31  -     16.0  fc3  ±  0.70 

In  order  to  vary  the  proceeding  as  much  as  possible  from 
that  of  the  former  investigation,  I  now  express  dc  in  terms  of 
dX  and  6fi,  which,  for  the  time  being,  I  take  as  the  corrections 
to  the  heliocentric  longitude  and  latitude  of  Venus  referred 
to  the  Earth,  and  these  again  in  terms  of  dv  and  sin  166, 
which  latter,  for  brevity,  I  call  u.  The  first  transformation  is 
made  with  the  coefficients  of  p.  71,  where  we  have  put  x  and 
—  y  for  6\  and  d/3,  and  the  last  by  the  equations 

// 

6X  =  6v  +  0.06  u 
8p  =  u  —  0.06  v 

Putting  Ui  for  the  value  of  u  in  1765,  we  have,  in  consequence 
of  the  known  change  in  the  motion  of  the  node, 

// 

In  1761;  u  =  Ui  +  0.11 
In  1769;  u  =  ^  —  0.11 


80]  MOTION   OF   THE  NODE   OF  VENUS.  161 

We  thus  have  the  four  equations  which  follow  for  determining 
6v  and  HI,  the  former  being  supposed  the  same  at  the  times  of 
the  two  transits. 

-  .84  6v  -  .55  M!  +  *a  =  +  0.15  -  2.2  &2  i  0.09 
+  .73        -  .69      +  z3  =  +  0.01  -  0.5  fc3  ±  0.03 

-  .69        +  .73      +  22  =  -  0.10  -  2.3  A"2  ±  0.04 
+  .81        +  .60      +  s3  =  +  0.10  -  1.3  &3  ±  0.06 

Eliminating  z.2  and  03  by  subtracting  the  first  equation  from 
the  third,  and  the  second  from  the  fourth,  we  have— 

.15  6v  +  1.28  %  =  -  o'.25  -~  o'.l  A;2  i  0.10 
.08  <*»  +  1.29  ui  =  +  0.09  -  0.8  fc3  ±  0.07 

We  thus  have  for  Ui  the  value 

m  =  -  07/.04  -  0.08  6v  -  0.03  A:2  -  0.36  fc3  i  Ox/.05 

dv  can  not  be  determined  independently  of  z2  and  «3.  Assum- 
ing these  quantities  to  be  equal,  we  have  already  found  it  to 
be  only  0/7.02,  and  may  therefore,  to  determine  its  probable 
effect  upon  the  result  by  assigning  to  it  the  value 


In  the  former  paper  I  have  found  for  k2  and  &3  the  values 

fc2  =  +  0.040  i  0.040 

Jc3  =  -  0.034  ±  0.040 

A  preliminary  correction  of  +  2/7.02  having  been  applied  to 
the  tabular  orbital  latitude,  we  have,  for  the  epoch  1765.5, 

sin  id  6  =  +  1".99  i  0".06 

Combining  this  result  with  that  of  the  transits  of  1874  and 
1882,  we  have  the  following  results,  which  are  compared  with 
those  of  meridian  observations  : 

// 
Transits  of  Yenus  alone    ......     sin  i  Dt  $6  =  —  2.82 

Meridian  observations  alone      ....  "  —  2.45 

Combined  solution     ........  «  _  2.71 

Adjusted  with  other*  results  (§46)  .     .     .  «  —2.73 

Adopted       ...........  «  —2.77 

5690  N  ALM  -  11 


162  MOTION  OF  THE  NODE  OF  VENUS.  [80 

The  adopted  result  is  the  one  which  seems  the  most  probable. 
For  the  final  probable  error  we  are  to  include  that  of  the  pre- 
cession and  of  the  Sun's  longitudes  at  the  two  epochs.  We 
may  estimate  the  combined  value  of  these  at  i  1",  correspond- 
ing to  an  error  of  0".06  in  sin  i  Dt  66.  Thus  we  have 

sin  i  Dt  66  =  —  2". 77  i  //.084 

I  conceive  this  mean  error  to  be  as  real  as  any  that  can  be 
determined  in  astronomy.  This  conviction  rests  upon  the  fact 
(1)  that  the  systematic  errors  affecting  the  four  contacts  are 
shown  to  be  small  by  the  general  minuteness  of  the  four  values 
of  dc;  (2)  that  whatever  systematic  errors  may  affect  the 
formation  or  disappearance  of  the  thread  of  light  are  almost 
completely  eliminated  from  the  mean  of  the  transits  of  1761 
and  1769  by  the  method  in  which  the  observations  have  been 
combined.  The  accordance  of  the  observations  of  external 
contact  made  at  the  same  transits  strengthens  this  view. 

The  equation  thus  derived  takes  the  place  of  the  sixth 
equation  of  §  63  and  should  have  twice  the  weight  there 
assigned.  As  the  mass  of  the  Earth  determined  by  the  secu- 
lar variations  rests  mainly  on  this  equation,  I  shall  first  con- 
sider it  alone.  Expressing  the  theoretical  secular  variation  of 
sin  i66  in  terms  of  the  above  observed  value,  we  find  that  the 
observed  motion  of  the  node  of  Yenus  gives  the  equation 

0".26  v  —  29". 2  v1  —  43".2  v"  =  +  0".48  i  0//.084         (a) 
which  gives  for  v"  the  value 

v"  =  —  0.0  ill  4-  0.006  v  -  0.676  v1  i  .0019 

The  value  of  the  solar  parallax  for  v"  —  0  is  8" .811.  Hence, 
for  the  value  expressed  in  terms  of  the  corrections  to  the 
assumed  masses  of  Yeuus  and  Mercury,  this  equation  gives 

n  =  8".778  +  0".020  r  —  1".98V 

We  have  found  from  the  periodic  perturbations 

//  // 

v  -  _  0.055    i  .25 
v1  =  +  0.0080  i  .0025 


80]  SOLAR  PARALLAX.  163 

Whence, 

//  // 

Y"  =  -  0.0168  i  .0029 
n  =      8.762    i  .0086 

This  result  of  observation,  errors  and  unknown  actions  aside, 
Fcan  not  suppose  to  be  affected  by  any  other  mean  error  than 
that  here  assigned. 

We  have  now  to  consider  how  far  this  result  may  be  recon- 
ciled with  the  others  by  changes  in  the  masses  of  Mercury 
and  Venus.  No  admissible  change  in  the  former  could  greatly 
affect  the  result.  The  question  then  arises  whether  the  dis- 
crepancy may  not  be  due  to  an  error  in  the  concluded  mass 
of  Venus.  In  making  so  large  a  change  in  this  element,  we 
meet  with  insuperable  difficulties.  The  observed  motion  of 
the  ecliptic,  which  is  a  fairly  well-determined  quantity,  indi- 
cates a  still  further  increase  of  this  mass.  We  may  put  this 
difficulty  in  another  form.  The  observed  motion  of  the  node 
of  Venus  is  a  relative  one,  consisting  in  the  combined  effect  of 
the  motion  of  the  ecliptic  around  an  axis  at  right  angles  to  the 
node  of  Venus,  and  an  absolute  motion  of  the  orbit  of  Venus 
around  nearly  the  same  axis.  This  motion  of  the  ecliptic 
depends  mainly  on  the  mass  of  Venus ;  the  absolute  motion 
Of  the  orbit  of  Venus  mainly  on  that  of  the  Earth.  If,  now,  we 
determine  the  motion  of  the  ecliptic  from  observation,  we  shall 
find  that  the  relative  motion  of  the  orbit  of  Venus  still  unac- 
counted for  is  yet  greater  than  we  have  supposed  it  to  be,  and 
should  therefore  find  a  yet  smaller  mass  of  the  Earth  than  that 
heretofore  concluded. 

The  determination  of  the  mass  of  Venus  already  made  from 
observations  of  the  Sun  and  Mercury  seems  to  admit  of  no 
doubt.  We  can  not  conceive  that  the  mean  of  fifteen  deter- 
minations, made  during  one  hundred  and  thirty  years,  at  dif- 
ferent observatories,  which  determinations  are  so  separated  as 
to  be  entirely  independent  of  each  other,  can  be  affected  by 
any  considerable  common  error.  The  entire  accordance  of  the 
result  thus  reached  from  the  periodic  perturbations  produced 
by  Venus  with  that  from  a  combination  of  all  the  secular 
variations,  as  shown  in  Chapter  VI,  strengthens  the  result 
yet  further.  Unknown  actions  and  possible  defects  of  theory 


164  SYSTEMATIC   ERRORS   OF   PARALLAX.  [tO,  M 

aside,  it  seems  to  me  that  the  value  of  the  solar  parallax 
derived  from  this  discussion  is  less  open  to  doubt  from  any 
known  cause  than  any  determination  that  can  be  made. 

Possible  systematic  errors  in  determinations  of  the  parallax. 

81.  We  have  now  to  return  to  the  other  values,  in  order  to 
see  to  what  extent  they  may  be  affected  by  systematic  error. 
I  have  already  excused  myself  from  discussing  the  validity  of 
the  assumed  relation  between  the  constant  of  aberration  and 
the  velocity  of  light,  because  there  is  nothing  valuable  to  be 
said  on  the  subject,  and  have  alluded  to  the  possible  sources 
of  systematic  error  in  the  Pulkowa  determinations  of  aberra- 
tion. It  is  worthy  of  attention  here  that  the  very  best  of  these 
determinations,  that  of  NYR^N  with  the  prime  vertical  transit, 
in  resp,ect  to  the  care  with  which  it  was  made,  and  the  general 
accordance  of  the  entire  work  throughout,  gives  a  result  most 
accordant  with  that  under  consideration.  In  fact,  to  the  value 
8". 77  of  the  solar  parallax  corresponds  the  value  20//.55  of 
the  constant  of  aberration,  which  is  larger  by  only  0//.02  than 
the  result  of  NYREN'S  best  determinations. 

A.S  for  miscellaneous  determinations  of  the  constant,  it  is  to 
be  remembered  that  the  corrections  applied  to  a  part  of  the 
separate  values  on  account  of  the  Chandlerian  inequality  of 
latitude  are  somewhat  doubtful,  and  the  general  mean  mav 
have  been  affected  by  a  few  huudredths  of  a  second  in  conse- 
quence. It  is  not,  however,  possible  to  determine  the  amount 
of  the  correction,  except  by  an  exhaustive  rediscussion  of  the 
whole  of  the  original  observations,  and  even  then  the  result 
would  still  be  doubtful. 

Next  in  the  order  of  weight  we  have  the  results  of  measures 
on  the  minor  planets  with  the  heliometer,  on  GILL'S  plan.  I 
have  already  remarked  upon  the  possible  error  in  such  obser- 
vations arising  from  the  probable  difference  of  color  between 
the  planet  and  the  star.  A  hypothetical  estimate  of  the 
amount  of  this  error  is  worth  attempting.  Let  us  assume  that 
in  the  case  of  a  minor  planet  the  mean  of  the  visible  spec- 
trum corresponds  to  the  line  D,  and  that  in  the  case  of  a  star 
the  same  mean  is  halfway  between  the  lines  D  and  E. 


81]  SYSTEMATIC   ERRORS   OF  PARALLAX.  165 

The  index  of  refraction  of  air  has  been  determined  inde- 
pendently by  KETTLER  and  LORENTZ  for  the  different  rays. 
The  mean  of  their  results  for  the  rays  D  and  E  is 

For  D,  n  =  1.000  2920 
ForE,  n  =  1.000  2940 

These  results  are  accordant  in  giving  a  dispersion  between 
these  two  lines  equal  to  about  .0037  of  the  total  refraction. 
We  have  hypothetically  taken  the  extreme  possible  difference  . 
between  planet  and  star  to  be  one-half  of  this.  „  At  an  altitude 
of  45°,  where  the  refraction  is  about  60",  the  error  would  be 
0".ll.  At  an  altitude  of  30°  the  error  would  be  0".20.  We 
are  thus  led  to  the  noteworthy  conclusion : 

If  the  difference  between  the  spectra  of  a  minor  planet  and  a 
comparison  star  is  such  that  the  means  of  their  respective  visible 
spectra,  or  the  apparent  amounts  of  their  respective  refractions, 
differ  by  one- tenth  of  the  space  between  D  and  E,  an  error  of 
0" .02  or  0" .03  may  be  produced  in  the  apparent  parallax  of  the 
planet. 

The  question  thus  arising  maybe  readily  settled  by  measures 
with  the  heliometer.  The  distances  of  pairs  of  stars  differing 
as  widely  as  possible  in  color  should  be  measured  at  different 
altitudes,  when  one  is  nearly  above  or  below  the  other,  in 
order  to  see  what  difference  of  refraction  depending  on  the 
color  is  indicated.  A  colored  double  star,  such  as  ft  Oygni, 
might  also  be  used  for  the  same  purpose. 

The  minor  planets  are  of  different  colors.  I  am  not  aware 
of  any  evidence  that  Victoria  or  Sappho  differ  in  color  from 
the  average  of  the  stars,  but  1  believe  that  Iris  is  somewhat 
yellow,  or  reddish.  Kow,  in  this  connection,  it  is  a  significant 
fact  that  the  parallax  found  from  observations  of  Iris,  8".82o, 
is  the  largest  by  GILL'S  method. 

I  have  already  remarked  that  the  value  of  the  solar  parallax 
derived  from  the  parallactic  equation  of  the  Moon  is  one  of 
which  the  probable  mean  error  is  subject  to  uncertainty. 
While  it  is  true  that  the  value  may  be  smaller  than  that  we 
have  assigned,  we  must  also  admit  that  it  may  be  much  larger. 

The  probable  error  of  the  determination  by  the  lunar  equa- 
tion of  the  Earth  is  larger  than  that  of  any  other  method.  At 


166  RESULTS  FOR  THE   SOLAR  PARALLAX.  [82 

the  same  time  I  do  not  think  that  it  is  liable  to  systematic 
error,  and  we  must  therefore  regard  the  mean  error  assigned 
as  real. 

Results  for  the  solar  parallax  after  making  allowance  for  prob- 
able systematic  errors. 

82.  Let  us  now  see  whether  we  can  reach  a  satisfactory 
result  by  making  a  liberal  allowance  for  the  more  or  less 
probable  sources  of  systematic  error  just  pointed  out.  The 
modifications  we  make  in  the  weights  formerly  assigned  are 
these:  We  reduce  the  weight  of  GILL'S  Ascension  result  to 
one-half,  owing  to  the  uncertainty  arising  from  the  color  of  the 
planet  Mars.  We  retain  the  Pulkowa  determinations  of  the 
constant  of  aberration  with  their  full  weight,  but  reduce  the 
weight  of  the  miscellaneous  determinations.  In  the  case  of 
the  parallactic  inequality,  we  reduce  the  weight  for  the  reasons 
already  given.  We  omit  Iris  from  the  determination  from  the 
minor  planets.  We  also  reduce  to  one- half  its  former  value 
the  relative  weight  assigned  to  measures  of  Venus  on  the  Sun, 
on  the  theory  that  the  actual  mean  error  must  be  larger  than 
that  given  by  the  discordance  of  results.  Our  combination 
will  then  be  as  follows : 

wt. 

From  the  motion  of  the  node  of  Venus  ....  n  =  8.708  10 
From  GILL'S  Ascension  observations  ....  8.780  1 

From  the  Pulkowa  constant  of  aberration  .     .     .  8.793     40 

From  contacts  of  Venus  with  the  Sun's  limb    .     .  8.794      3 

From  heliometer  observations  on  Victoria  and 

Sappho 8.799      5 

From  the  parallactic  inequality  of  the  Moon   .     .  8.794    10 
From  miscellaneous  determinations  of  the  con- 
stant of  aberration 8.806    10 

From  the  lunar  inequality  in  the  motion  of  the 

Earth 8.818      1 

From  measures  on  Venus  in  transit 8.857       1 

Mean  result,  ignoring  the  first ;  8".7965i  .0045 

This  mean  result  still  differs  from  that  given  by  the  motion 
of  the  node  of  Venus  by  more  than  five  times  the  probable 
error  of  the  latter,  and  is  yet  farther  from  the  combined  result 


82]        RESULTS  FOR  THE  SOLAR  PARALLAX.        167 

of  all  the  secular  variations,  so  that  no  reconciliation  is  brought 
about. 

The  embarrassing  question  which  now  meets  us  is  whether 
we  have  here  some  unknown  cause  of  difference,  or  whether 
the  discrepancy  arises  from  an  accidental  accumulation  of 
fortuitous  errors  in  the  separate  determinations.  We  have 
already  discussed  the  former  hypothesis,  and  have  been  unable 
to  find  any  reasonably  probable  cause  of  abnormal  action. 
The  motion  of  the  planes  of  the  orbits  is  that  which  is  least 
likely  to  deviate  from  theory,  because  it  is  independent  of 
all  forms  of  action  depending  upon  distance  from  the  Sun, 
or  upon  the  velocity  of  the  planet. 

An  examination  and  comparison  of  all  the  results  shows  one 
curious  feature:  the  unanimity  with  which  the  secular  varia- 
tions speak  against  the  large  value  of  the  solar  parallax,  or 
of  the  mass  of  the  Earth,  as  the  one  quantity  at  fault.  The 
adopted  motion  of  the  node  of  Venus  is  sustained  not  only  by 
the  meridian  observations,  but  by  the  external  contacts  at  the 
transits  of  1761  and  1769,  and,  weakly,  by  a  comparison  of  the 
transits  of  1874  and  1882. 

If  we  determine  the  correction  of  the  mass  of  the  Earth  from 
other  secular  variations  than  that  of  the  node  of  Venus,  by 
the  equations  of  §  63,  we  have,  after  eliminating  the  masses  of 
Mercury  and  Venus, 

v"  =  -0.029;  p.  e.  ±  .018 
If,  instead  of  eliminating  these  values,  we  put 

v  =  +  .08;  v1  =  +  .0080; 
we  have 

v"  =  -0.026;  p.  e.  i  .014 

In  each  case  the  value  of  the  parallax  is  yet  smaller  than  that 
found  from  the  motion  of  the  node  of  Venus.  I  have  already 
remarked  that  the  observed  motion  of  the  ecliptic  indicates 
an  increase  of  the  mass  of  Venus. 

The  question  thus  takes  the  form,  whether  it  is  possible  that 
the  mean  of  the  eeven  determinations  of  the  solar  parallax 

TT  =  8".797  i  ".0035 


168  DEFINITIVE  ADJUSTMENT.  [82,  83 

can  with  reasonable  possibility  be  in  error  by  aii  amount  the 
correction  of  which  would  bring  it  within  the  range  of  adjust- 
ment of  the  other  quantities. 

From  what  has  already  been  said  of  the  systematic  errors 
to  which  every  one  of  the  determinations  may  be  liable,  it  is 
evident  that  we  should  have  no  difficulty  in  accepting  the 
necessary  reduction  of  each  of  the  separate  values.  The 
improbability  which  meets  us  is  not  so  much  the  amount  of 
the  individual  errors  of  the  determinations  as  the  fact  that 
seven  of  the  eight  independent  determinations  should  all  be 
largely  in  error  in  the  same  direction.*  Still,  under  the  cir- 
cumstances, we  must  admit  this  possibility,  and  make  what 
seems  to  be  the  best  adjustment  of  all  the  results. 

Definitive  adjustment. 

83.  In  making  the  definitive  adjustment  I  shall  proceed  on 
the  supposition  that  no  correction  is  necessary  to  the  adopted 
mass  of  Mars.  I  also  go  on  the  principle  that  no  result  is  to 
be  rejected  on  account  of  doubt  or  discordance,  except  when 
it  is  affected  with  a  well-established  cause  of  systematic  error, 
and  shows  a  large  deviation  in  the  direction  in  which  this 
cause  would  act.  At  the  same  time  it  will  be  admissible  to 
diminish  the  weights  in  special  cases,  on  account  of  causes  of 
systematic  error  which  we  know  to  exist,  although  we  can  not 
determine  the  directions  in  which  they  would  act ;  and  also  on 
account  of  deviations  so  wide  as  to  show  that  the  probable 
error  of  the  result  must  have  been  greatly  underestimated. 
Proceeding  on  this  plan,  we  might  reweight  the  last  eight 
results  for  the  solar  parallax,  so  as  to  get  a  result  slightly 
different  from  8". 797.  But  1  doubt  whether  such  a  reweight- 
ing  would  not  involve  an  objectionable  bias. 

We  might  diminish  the  weight  of  the  result  given  by  the 
Pulkowa  constant  of  aberration  on  the  ground  that  no  one 
method  should  have  so  preponderating  a  weight  as  this  has. 
If  we  did  so  the  result  might  be  increased  to  8".800.  We 

*  For  a  very  searching  criticism  of  the  systematic  errors  with  which  the 
determinations  of  the  solar  parallax  may  be  affected,  reference  may  be 
made  to  the  first  two  articles  by  Dr.  DAVID  GILL,  in  Vol.  I  of  The  Observa- 
tory. 


83]  DEFINITIVE   ADJUSTMENT.  169 

might  very  largely  increase  the  relative  weight  assigned  to 
the  heliometer  observations  on  Victoria  and  Sappho,  but  no 
admissible  increase  would  appreciably  change  the  result.  We 
might  also  diminish  the  relative  weight  of  the  largely  dis- 
cordant result  derived  from  measures  of  Venus  during  transit. 
But  as,  by  throwing  out  this  result  altogether,  we  should  only 
diminish  the  mean  by  ".001,  it  is  scarcely  worth  while  to  do 
so.  Altogether  no  rediscussion  of  the  relative  weights  seems 
necessary. 

On  the  other  hand,  the  weight  which  we  assign  to  the  mean 
result  will  enter  as  a  very  important  factor  into  the  final 
adjustment.  This  is  a  point  on  which  it  is  impossible  to  reach 
a  positive  numerical  conclusion  by  any  mathematical  process. 

If,  as  one  extreme  case,  we  consider  that  the  mean  error  of 
each  separate  result  corresponds  to  i07/.03  for  weight  unity, 
we  shall  have  a  mean  error  of  rt".0035  for  the  value  8". 797. 
The  result  will  not  be  very  different  if  we  determine  the  mean 
error  from  the  discordance  of  the  eight  separate  results.  On 
the  other  hand,  if  we  include  the  deviation  of  the  result  given 
by  the  motion  of  the  node  of  Venus,  the  mean  error  for  weight 
unity  will  be  increased  to  i  0".0045.  The  latter  is  undoubt- 
edly the  most  logical  course,  so  long  as  we  proceed  on  the 
hypothesis  that  the  deviations  of  the  final  adjustment  can  all 
be  explained  as  due  to  fortuitous  errors.  If  we  include  a  com- 
parison with  the  results  of  all  the  secular  variations  we  shall 
have  a  yet  larger  mean  error.  To  show  the  result  of  assigning 
one  weight  or  the  other  I  shall  make  two  solutions,  A  and  B, 
in  one  of  which  a  less  and  in  the  other  a  greater  weight  will 
be  assigned. 

To  the  value  8".797  i  .005  or  ±  .007  of  the  solar  parallax 
corresponds 

r"  =  -  0.049  i  .0016  or  ±  .0025 

According  as  we  assign  one  weight  or  the  other  to  this  result, 
we  may  take  as  the  corresponding  equation  of  condition  of 
weight  unity 

(A):  400^' =-2.0 

°r     (BH  600,"  =  -2.9  W 


170  DEFINITIVE  ADJUSTMENT.  [83 

The  masses  of  Venus  and  Mercury,  determined  by  methods 
independently  of  the  secular  variations,  also  enter  as  conditions 
into  the  adjustment.  I  have,  however,  made  a  revision  of  the 
preliminary  adjustment  given  in  §  64,  the  latter  being  based  on 
the  results  of  §§  32-38;  whereas  it  is  better  to  use  the  defini- 
tive results  of  the  combination  used  in  §  46. 

For  the  mass  of  Mercury  the  result  found  in  §  53  by  the 
last  combination  is 


The  values  of  the  denominator  corresponding  to  the  mean 
limits  here  assigned  are 

5  890  000  and  12  210  000 

These  limits  are  so  wide  as  to  include  all  admissible  results  for 
the  mass  of  Mercury.  Moreover,  we  can  not  definitely  say  that 
the  value  (6)  of  this  mass  is  markedly  greater  or  less  than  that 
given  by  the  weighted  mean  of  all  other  results,  since  we 
might  so  weight  the  latter  as  to  give  a  result  greater  or  less 
without  transcending  the  bounds  of  judicious  judgment.  I 
conceive,  therefore,  that  we  are  justified  in  reducing  the  mean 
error  to  i  0.26,  which  will  give  as  the  equation  of  condition 

r=  -  0.055  i  0.25 
and  hence 

40  x  =  -  0.22  i  1  (c) 

When,  in  the  normal  equation  for  the  mass  of  Venus,  given 
by  the  observations  on  Mercury,  we  substitute  the  values  of 
the  secular  variations  found  from  the  general  combination  of 
§  46,  the  result  is 

v1  =  —  0.0114 

Combining  this  with  the  result  from  the  Sun,  we  have 

v1  =  -  0.0117 

In  view  of  the  fact  that  the  mass  derived  from  observations  of 
Mercury  may  be  affected  by  systematic  errors  of  the  kind 


83]  DEFINITIVE  ADJUSTMENT.  171 

shown  and  discussed  in  §  53,  the  mean  error  formerly  assigned 
to  this  result  should  be  somewhat  diminished.     The  result  is 


406  600 
From  this  we  have 

v'  =  +  0.0084  ±  .0030 

For  the  equation  of  condition  of  weight  unity  I  take 

330  v'  =  +  2.8  (d) 

With  these  equations  of  condition  we  have  to  combine  the 
eleven  equations  of  §63,  which  we  use  unchanged,  except  that 
we  double  the  weight  assigned,  to  the  sixth  equation,  that 
derived  from  the  motion  of  the  node  of  Venus,  on  account  of 
the  smaller  probable  error  of  the  result  of  our  preceding  redis- 
cussion,  and  use  the  value  of  the  absolute  term  found  in  §80. 

If  we  accept  the  view  that  all  the  perihelia  move  according 
to  the  same  law  of  gravitation  toward  the  Sun,  namely,  that 
expressed  by  HALL'S  hypothesis,  then  the  value  of  the  quan- 
tity 6  in  the  formula  expressing  the  law  of  gravitation  is  so 
well  determined  by  the  motions  of  Mercury  that  it  becomes 
legitimate  to  use  the  observed  motions  of  the  perihelia  of  the 
other  three  planets  as  equations  of  condition.  But  since  it  is 
not  impossible  that  the  minor  planets  between  Mars  and 
Jupiter  may  have  an  appreciable  influence  on  the  motion  of 
the  perihelion  of  Mars,  it  is  a  question  whether  we  should  not 
exclude  that  motion  from  the  equations. 

The  conditional  equations  given  by  the  motions  of  the  three 
perihelia  in  question  are  found  by  comparing  the  results  of 
§  §  46,  54,  and  61.  They  are 

40  x  +    0  v1  +  20  v"  =  +  1.0 
-14+46      +0        =  -  0.3  (e) 

2       -  13       +61        =  +  0.7 

The  conditional  equations  to  be  combined  are  the  eleven 
equations  of  §63,  the  sixth  of  which  is  to  have  double  weight^ 
and  the  six  equations  (a),  (c),  (d),  and  (e). 


172 


DEFINITIVE   ADJUSTMENT. 


[83 


The  normal  equations  to  which  we  are  thus  led  are  the 
following,  which  show  the  results  of  the  four  combinations  we 
may  make  according  as  we  use  (A)  or  (B)  for  the  equation 
given  by  the  mass  of  the  Earth,  and  omit  or  include  the  third 
equation  (d),  which  is  given  by  the  motion  of  the  perihelion 
of  Mars. 

(a.)  Including  the  motion  of  the  perihelion  of  Mars. 


9  607,r  —      7  147 7';  —    11  335j/"  =  +  220 

=  -587 
=  -  3388  (A) 
=  -  4328  (B) 


7  147  +  267  174  +  168  727 
11  335  +  168  727  +  406  300 
1 1  335  ,  +  168  727  +  606  300 


(/?.)  Omitting  the  motion  of  the  perihelion  of  Mars. 

9  603#  -      7  12lv'  -    11  457 v"  =  +  219 

-  7  121     +  267  003      +  169  520       =  -  578 

-  11  457     +  169  520      +  402  578       =  -  3431  (A) 

-  11  457     +  169  520      +  602  578      =  -  4371  (B) 

The  results  of  the  solutions  in  the  four  cases  are: 


Aa 

A// 

B« 

B/? 

X     -f 

0.0147 

+  0.0142 

+  0.0161 

+  0.0158 

V     -j- 

0.147 

+  0.142 

-f  0.161 

+  0.158 

V1    + 

0.004  34 

+  0.004  60 

+  0.003  10 

+  0.003  i>.") 

v"  — 

0.009  73 

—  0.01005 

-  0.007  70 

—  0.007  87 

1 

-V-  m 

6  539  000 

6  567  000 

6  460  000 

6  477  000 

1 

+  m' 

408  230 

408  120 

408  730 

408  670  , 

7T 

8".783 

8//.782 

8".789 

8".788 

I  conceive  that  if  the  secular  variations,  especially  the  motion 
of  the  node  of  Venus,  are  not  affected  by  any  unknown  cause, 
some  mean  between  these  should  be  regarded  as  the  most 
probable  solution.  The  result  does  not,  however,  bring  about 
a  satisfactory  reconciliation.  We  still  find  ourselves  confronted 
by  this  embarrassing  dilemma:  Either  there  is  something 
abnormal  in  connection  with  the  node  of  Venus,  due  to  an 
unknown  cause  acting  on  the  planet,  to  some  extraordinary 
errors  in  the  observations  or  their  reduction,  or  to  some  error 
in  the  theory  on  which  the  discussion  is  based,  or  the  deter- 


83, 84,  85]          ADOPTED    PARALLAX   AND   MASSES.  173 

ruinations  of  the  solar  parallax  are  nearly  all  in  error  in  one 
direction  by  amounts  which  are,  in  more  than  one  case,  quite 
surprising. 

Possible  causes  of  the  observed  discordances. 

84.  Two  possible  causes  of  discordance  may  be  suggested, 
one  of  which  has  not  been  touched  upon  at  all  in  the  preceding 
chapters,  and  one  perhaps  inadequately.    As  to  the  hypothesis 
of  non-sphericity  of  the  Sun,  considered  in  §56,  it  may  be 
remarked  that  Dr.  HARTZER  shows  that  an  ellipticity  of  the 
Sun  sufficient  to  produce  the  observed  motion  of  the  perihelion 
of  Mercury  would  cause  a  direct  motion  of  5".l  in  the  motion 
of  the  node  of  Venus.    This  would  correspond  to  a  change  of 
0".30  in  the  value  siniDt#  and  would  therefore  go  far  toward 
reconciling  the  discrepancy.    But  it  is  easy  to  see  that  this 
cause  would  produce  a  secular  motion  of  —  2".6  in  the  inclina- 
tion of  Mercury.    We  have  seen  that  the  observed  motion  of 
the  inclination  already  exceeds  the  theoretical  motion  by  0".38; 
so  that  introducing  the  hypothesis  of  ellipticity  of  the  Sun  we 
should  have  a  discrepancy  of  about  S^.O  between  theory  and 
observation.    This  conclusion  alone  seems  fatal  to  the  theory, 
which  otherwise  has  been  shown  to  be  scarcely  tenable. 

The  other  possible  cause  is  an  inequality  of  long  period ; 
especially  one  depending  on  the  argument  \3l" — 81'  which 
has  a  period  of  about  two  hundred  and  forty  three  years.  A 
very  simple  computation  shows  that  the  coefficient  of  this  term 
is  only  of  the  order  of  magnitude  (V'.Ol. 

It  is  a  curious  coincidence  that  if  we  had  neglected  to  add 
the  mass  of  the  Moon  to  that  of  the  Earth,  in  computing  the 
secular  variations,  the  discrepancy  would  not  have  existed. 

Adopted  values  of  the  doubtful  quantities. 

85.  The  practical  question  which  has  been  before  the  writer 
in  working  out  the  preceding  results  is :  What  values  of  the 
constants  should  be  used  in  the  tables  of  the  celestial  motions 
of  which  the  results  of  this  discussion  are  to  form  the  basis  ? 
Should  we  aim  simply  at  getting  the  best  agreement  with  obser- 
vations by  corrections  more  or  less  empirical  to  the  theory  ? 
It  seems  to  me  very  clear  that  this  question  should  be  answered 
in  the  negative.    No  conclusions  could  be  drawn  from  future 


174  ADOPTED   PARALLAX   AND   MASSES.  [85 

comparisons  of  such  tables  with  observations,  except  after 
reducing  the  tabular  results  to  some  consistent  theory.  The 
imposition  of  such  a  labor  upon  the  future  investigator  is  not 
to  be  thought  of.  Moreover,  there  is  no  certainty  that  the 
tables  which  would  best  represent  past  observations  would 
also  best  represent  future  ones.  Our  tables  must  be  founded 
on  some  perfectly  consistent  theory,  as  simple  as  possible,  the 
elements  of  which  shall  be  so  chosen  as  best  to  represent  the 
observations. 

In  choosing  the  theory  and  its  constants  we  have  again  a 
certain  range.  If  we  accept  the  necessity  of  assuming  the 
secular  variations  of  the  orbits  of  Mercury  and  Venus  to  be 
affected  by  the  action  of  unknown  masses  of  matter,  then  the 
simplest  course  to  adopt  is  to  construct  our  theory  on  the  sup- 
position of  a  planet  or  group  of  planets  between  Mercury  and 
Venus. 

It  seems  to  me  that  the  introduction  of  the  action  of  such  a 
group  into  astronomical  tables  would  not  be  justifiable.  The 
more  I  have  reflected  upon  the  subject  the  more  strongly 
seems  to  me  the  evidence  that  no  such  group  can  exist,  and, 
indeed,  that  whatever  anomalies  exist  can  not  be  due  to  the 
action  of  unknown  masses  of  matter. 

Besides,  the  six  elements  of  such  a  group  would  constitute 
a  complication  in  the  tabular  theory. 

On  the  other  hand,  it  did  not  seem  to  me  best  that  we  should 
wholly  reject  the  possibility  of  some  abnormal  action  or  some 
defect  between  the  assumed  relations  of  the  various  quanti- 
ties. What  I  finally  decided  on  doing  was  to  increase  the  theo- 
retical motion  of  each  perihelion  by  the  same  fraction  of  the 
mean  motion,  a  course  which  will  represent  the  observations 
without  committing-  us  to  any  hypothesis  as  to  the  cause 
of  the  excess  of  motion,  though  it  accords  with  the  result  of 
HALL'S  hypothesis  of  the  law  of  gravitation ;  to  reject  entirely 
the  hypothesis  of  the  action  of  unknown  masses,  and  to  adopt 
for  the  elements  what  we  might  call  compromise  values  between 
those  reached  by  the  preceding  adjustment  and  those  which 
would  exist  if  there  is  abnormal  action.  The  exigency  of  hav- 
ing to  prepare  the  tables  required  me  to  reach  a  conclusion  on 
this  subject  before  the  final  revision  of  the  preceding  discus- 


85,86]  FUTURE  DETERMINATIONS.  175 

sion,  so  that  the  numbers  used  are  not  wholly  based  upon  it. 
The  conclusions  I  have  reached  are  these: 

Since,  if  there  is  nothing  abnormal  in  the  theory,  the  solar 
parallax  is  probably  not  much  larger  than  8".780,  and  if  there 
is  anything  abnormal  it  is  probably  as  large  as  8".795  or  even 
8' '.800,  we  may  adopt  the  value  8". 790  as  one  which  is  almost 
certainly  too  large  on  the  one  hypothesis  and  too  small  on  the 
other,  and  which  is  therefore  best  adapted  to  afford  a  decision 
of  the  question. 

For  the  mass  of  Venus  I  took,  as  an  intermediate  value, 

m'  =1-1-408000 
For  the  mass  of  Mercury  I  took 

1  4-  6,000,000 

Actually  it  seems  that  this  mass  is  larger  than  the  most  prob- 
able one  on  either  hypothesis,  though  not  without  the  range  of 
easy  possibility. 

With  these  values  the  outstanding  difference  between  theory 
and  observation  in  the  centennial  motion  of  the  node  of 
Venus  is 

A  sin  i  Dt  0  =  0".25 

If  this  difference  arises  wholly  from  the  error  of  the  theory, 
then  between  the  transits  of  1874  and  2004  the  accumulated 
error  would  amount  to  0".32  in  the  heliocentric  latitude,  and 
about  0".8  in  the  geocentric  latitude.  Unless  an  improvement 
is  made  in  the  method  of  determining  the  position  of  Venus 
by  observation,  the  twentieth  century  must  approach  its  end 
before  this  difference  can  be  detected. 

Bearing  of  future  determinations  on  the  question. 

86.  The  following  shows  the  influence  which  subsequent 
determinations  of  the  principal  elements  will  have  upon  our 
judgment  as  to  the  solution  of  the  dilemma.  The  changes  in 
the  second  column  will,  by  emphasizing  the  discordance 
between  the  results,  tend  to  confirm  the  hypothesis  of  an 
abnormal  defect  in  the  theory,  while  the  opposite  ones,  in  the 
last  column,  will  tend  to  reconcile  theory  and  observation : 


USUVBRSITf 


176 


FUTURE   DETERMINATIONS. 


[86 


Element  or  quantity. 

Change  tending  to 
confirm  the  dis- 
cordance 
between  theory 
and  observation. 

Change  tending  to 
reconcile  exist- 
ing  theory  with 
observation. 

The  solar  parallax. 

Increase. 

Diminution. 

Longitude  of  the  node  of  Mercury. 

Increase. 

Diminution. 

Longitude  of  the  node  of  Venus. 

Increase. 

Diminution. 

Constant  of  aberration. 

Diminution. 

Increase. 

Mass  of  Venus. 

Increase. 

Diminution. 

Mass  of  Mercury. 

Diminution. 

Increase. 

Secular  diminution  of  the  obliquity. 

Diminution. 

Increase. 

Among  these  constants  are  some  the  values  of  which  can 
scarcely  be  decisively  obtained  except  by  observations  con- 
tinued through  half  a  century,  or  even  through  the  whole 
twentieth  century,  unless  improvements  are  made  in  our  pres- 
ent methods  of  observing. 

The  improvement  of  others,  however,  is  quite  within  the 
reach  of  the  astronomy  of  the  present  time.  Among  these 
the  constant  of  aberration  and  the  solar  parallax  have  the 
first  place.  The  more  accurate  determination  of  these  quanti 
ties  thus  assumes  an  importance  which  may  justify  some  sug- 
gestions on  the  subject. 

The  observations  made  on  the  European  continent  for  the 
detection  and  study  of  the  variations  of  latitude  have  been 
executed  with  such  precision  that  we  might  look  to  them  for  a 
marked  improvement  in  the  determination  of  the  constant  of 
aberration,  were  it  not  for  a  single  circumstance.  In  the  gen- 
eral average  few  are  made  after  midnight,  while  the  maxima 
and  minima  of  aberration  occur  in  the  morning  and  evening. 
The  extension  of  the  system  into  the  early  morning  therefore 
seems  desirable.  Although  these  observations  may  scarcely 
equal  in  accuracy  those  made  by  NYREN,  with  the  prime 


86]  FUTURE   DETERMINATIONS.  177 

vertical  transit,  they  have  the  advantage  of  not  requiring  so 
long  a  period  for  a  complete  observation.  The  great  disad- 
vantage of  the  prime  vertical  instrument  is  that  unless  a  star 
culminates  within  a  few  minutes  of  the  zenith,  an  hour,  or 
even  several  hours,  will  be  required  for  the  completion  of  a 
determination,  which  may  thus  be  made  impossible  by  the 
'advent  of  daylight.  It  may  be  remarked  in  this  connection 
that  the  northern  latitudes  of  the  European  observatories  are 
favorable  to  the  determination  of  the  aberration-constant. 

LOEWY'S  method  has  over  all  others  the  great  advantage  of 
being  independent  of  the  direction  of  the  vertical.  But  its 
application,  and  the  reduction  of  the  observations  made  with, 
it,  are  laborious  in  a  high  degree. 

So  far  as  practical  astronomy  has  yet  developed,  the  best 
method  of  directly  measuring  planetary  parallax,  and  there- 
fore the  only  one  to  be  considered,  is  that  of  GILL.  It  there- 
fore seems  desirable  that  measures  by  this  method  should  be 
continued.  At  the  same  time  it  is  very  necessary  that  the 
spectra  of  the  small  planets  to  be  used  should  be  carefully 
studied  photometrically,  and  that  the  probable  influence  of 
coloration  upon  the  measures  should  be  investigated. 

The  necessity  of  completing  the  present  work,  and  of  pro- 
ceeding immediately  to  the  construction  of  tables  founded 
upon  the  adopted  elements,  prevent  the  author's  awaiting  the 
mature  judgment  of  astronomers  upon  the  embarrassing  ques- 
tions thus  raised.  The  regret  with  which  he  accepts  this 
necessity  is  weakened  by  the  consideration  that  even  if  the 
solar  parallax  which  he  has  adopted  requires  the  largest  cor- 
rection to  which  it  can  reasonably  be  supposed  subject,  namely, 
one  of  — 0".015,  reducing  the  value  of  this  constant  to  8". 775, 
the  effect  of  the  error  will  not  be  prejudicial  to  the  astronomy 
of  the'immediate  future. 

More  important  will  be  the  error  0/x.035  in  the  constant  of 
aberration.  Yet  a  long-continued  series  of  observations  will 
be  necessary  to  establish  even  the  existence  of  such  an  error, 
and  should  it  prove  detrimental  in  any  astronomical  work  the 
evil  will  be  easily  remedied  by  a  slight  correction. 
5690  N  ALM 12 


CHAPTER  IX. 

DERIVATION  OF  RESULTS. 

Ulterior  corrections  to  the  motions  of  the  perihelion  and  mean 
.   longitude  of  Mercury. 

87.  In  §§32  and  46  we  have  reached  three  values  of  the 
correction  to  the  tabular  motion  of  the  perihelion  of  Mercury. 
Of  these  the  first  rests  on  meridian  observations  alone,  the 
second  on  the  combination  of  meridian  observations  with  trans- 
its, and  the  third  is  derived  by  substituting  in  the  eliminating 
equations  the  corrections  to  the  solar  elements  and  their  secular 
variations  which  result  from  observations.  The  three  values 
thus  reached  are  —  9".54,  — 1".01,  and  +  6".34.  The  pro- 
gressive divergence  of  these  values,  taken  in  connection  with 
the  discrepancy  pointed  out  in  §33,  leads  us  to  distrust  the 
influence  of  the  meridian  observations  upon  the  motion  of  the 
perihelion.  Under  these  circumstances  I  deem  it  advisable  to 
make  such  final  corrections  to  the  motions  in  mean  longitude 
and  mean  anomaly  as  will  best  satisfy  all  the  observed  transits 
over  the  disk  of  the  Sun.  In  doing  this  I  am  enabled  to  intro- 
duce the  results  of  a  preliminary  discussion  of  the  transits  of 
1891  and  1894.  By  combining  the  observations  of  these  two 
transits  with  those  of  the  older  ones  I  derive  the  following 
values  of  the  functions  Y  and  W  defined  in  §  31 : 

//          // 
Y  = -1.93- 3.03  T 

W= +  1.50  + 2.04  T 

The  preliminary  theory,  so  far  as  yet  investigated,  gives  for 
the  values  of  this  quantity, 

//          // 
Y  =  -  2.44  —  3.40  T 

W  =  +  1.38  +  1.3GT 
178 


87,  88]  PERIHELION  OF  MERCURY.  179 

Equating  these  values  to  the  corresponding  linear  functions 
of  the  corrections  to  /,  TT,  and  their  secular  motions,  we  have 
the  equations, 

//          // 

0.72  SI  +  0.28  67f  =  +  0.12  +  0.68  T 
+  1 .49      -  0.49       =  +  0.51  +  0.37  T 

We  find,  from  these  equations, 

//          // 

61  = +0.26  +  0.56  T 
Sn  =  -  0.24  +  0.97  T 

The  preliminary  values  to  which  these  corrections  are  appli- 
cable are 

//          // 

61  =  +0.04- 1.33  T 
6jr=  +  5.83  +  6.34  T 

The  definitive  values  thus  become 


61  = +  0.30- 0.77  T 

tf  TT  =  +  5.59+ 7.31  T 

Definitive  elements  of  the  f out  inner  planets  for  the  epoch  1850,  as 
inferred  from  all  the  data  of  observation. 

88.  We  have  made  a  fourth  solution  of  the  normal  equations 
which  give  the  corrections  to  the  elements  of  each  planet  by 
substituting  in  those  equations  the  definitive  values  of  all  the 
other  quantities,  including  the  values  of  the  secular  variations 
derived  from  theory.  In  making  this  substitution  for  Mercury, 
however,  the  ulterior  corrections  just  found  were  not  applied. 
The  values  of  the  unknowns  resulting  from  this  solution  are 
shown  in  the  first  column  of  the  next  table.  From  these 
numbers  are  derived  the  definitive  elements  for  1850, 'by  the 
following  processes: 

(a.)  By  multiplying  the  unknowns  by  the  appropriate  factor 
given  in  §  27,  we  have  the  corrections  of  the  tabular  elements 
at  the  mid- epoch  of  observations  for  each  planet.  These  cor- 
rections are  found  in  the  second  column. 

(/?.)  The  preceding  corrections  are  to  be  reduced  from  the 
respective  mid-epochs  to  1850.  This  reduction  is  found  by 


180  DEFINITIVE   QUANTITIES.  [88 

multiplying  the  definitive  correction  to  the  tabular  secular 
variation  by  the  elapsed  interval,  and  is  shown  in  the  third 
column. 

(y)  We  next  have  the  value  of  the  tabular  elements  for  the 
fundamental  epoch  1850,  January  0,  Greenwich  mean  noon. 
These  numbers  are  those  of  LEVERRIER'S  tables,  with  the 
following  modifications: 

(d)  The  reduction  from  1850,  January  1,  Paris  noon,  to 
January  0,  Greenwich  noon 

(f)  The  corrections  to  LEVERRIER'S  values  of  the  eccen- 
tricity and  perihelion  which  are  necessary  to  represent  those 
terms  in  the  perturbations  of  the  mean  longitude  which  depend 
only  upon  the  sine  and  cosine  of  the  mean  anomaly.  The 
theory  is  more  symmetrical  in  form  when  all  such  terms  are 
included  with  those  of  the  elliptic  motion.  In  LEVERRIER'S 
tables  they  have  the  following  values: 


Mercury  5  6v  =  4-  0.030  sin  I  -  0.111  cos  I 
Venus;  +0.010          +0.037 

Earth;  —0.067  -0.098 

Mars;  +1.061          +0.718 

These  terms  of  the  longitude  may  be  represented  by  the  follow- 
ing corrections  to  the  elements: 


Mercury;  de  =  +  0.058  dn  r=  0.0 

Venus;  -0.012  +2.3 

Earth;  +0.054  +1.4 

Mars;  +0.613  -1.0 

Applying  these  corrections  d  and  e  to  LEVERRIER'S  tabular 
quantities,  we  have  the  values  of  the  tabular  elements  as  given 
in  the  fourth  column.  Then  applying  the  preceding  correc- 
tions we  have  the  definitive  values  given  in  the  last  column. 

In  some  cases  this  derivation  is  modified.  Instead  of  using 
the  correction  to  the  perihelion,  mean  longitude  and  mean 
motion  of  Mercury  given  by  the  unknown  quantities  of  the 


88]  ELEMENTS  FOR  1850.  181 

equations,  we  have  used  the  values  for  1850  derived  from  the 
discussion  of  the  preceding  section. 

The  quantities  which  give  the  position  of  the  node  and 
inclination  have  been  treated  in  the  same  way  as  their  secular 
variations.  The  symbols  J  and  N  indicate  values  of  the 
unknown  quantities  related  to  the  corrections  of  the  elements 
J  and  N.  These  unknowns  are  then  changed  to  corrections  of 
the  elements  by  the  factors  of  §  27,  and  these  again  to  correc- 
tion of  the  inclination  and  node  by  the  equations  of  §41. 

In  the  case  of  the  node  of  Venus  two  values  are  given.  The 
value  (a)  is  that  which  follows  immediately  from  the  normal 
equations.  If  we  carry  forward  the  position  of  the  node  just 
derived  to  the  mean  epoch  of  the  last  two  transits  of  Venus, 
we  find  a  discrepancy  amounting  to  2".04  in  the  longitude, 
corresponding  to  a  difference  of  0".121  in  the  heliocentric  lati- 
tude. This  is  considerably  larger  than  the  probable  error  of 
the  results  of  the  observations  of  the  transits.  It  may,  there- 
fore, be  questioned  whether  the  latter  are  not  entitled  to  a 
greater  relative  weight  than  that  assigned,  owing  to  the  prob- 
able systematic  errors  of  the  meridian  observations.  A  second 
value  (b)  has  therefore  been  derived  from  the  observations  of 
the  transits  alone.  In  subsequent  investigations  we  may 
choose  between  these  two  values. 

Formation  of  definitive  elements  of  the  four  inner  planets,  for  tlit\ 
epoch  1850 7  January  0,  Oreemvich  mean  noon. 

Mercury. 

Unknown  of     Corr.  of      Red.  to  Tabular  Concluded 

equations.      element.       1850.  element.  element. 

//  //  //  // 

n    -.0940     -    0.77         0.0      538106654.49    538106653.72 

e    -  .0741     -    0.222  -  0.005  42  409.088          42  408.861 

n  +  .6763     -1-  5.59            0  7°5  7  13.78  7°5  7  19'!s7 

t  —.0402     +  0.30            0  323  11  23.53  323  11  23.83 

i  -.2762J--  0.64  -  0.07  7  0  7.71  7  0  7.00 

d  -.0001N+  3.88  -  0.27  46  33  8.63  46  33  12.24 


182  DEFINITIVE   QUANTITIES.  [88,89 

Formation  of  definitive  elements,  etc. — Continued, 
Venus. 

Unknown  of     Corr.  of      Ked.  to  Tabular  Concluded 

equations,      element.        1850.  element.  element. 


n  -  .1783  -  3.57 


0   210669165.04  210669161.47 


e  +  .1463 


0.439  -  0.165 


1  411.522 


1  411.796 


129 


7t  +  .0835  +  36.6  —16.4 

z  _  .1330  -  0.67  +  0.46.  243 

i  +  .0968  J  +  0.31  +  0.12    3 

0(a)+  .0126 N-  9.39  +  6.63   75 

0(b)        -20.36  +15.56 

Earth. 


27 
57 
23 
19 


14.3  129 
44.34  243 
34.83  3 
52.21  75 
75 


27 
57 
23 
19 
19 


34.5 

44.13 

35.26 

49.45 

47.41 


1.10 
0.12 

2.4 

0.15 

0.02 


129  602  767.84  129  602  766.74 


3  459.334 

100  21  43.4 
23  27  31.83 

99  48  18.72 


Mars. 


-  .1094  -  0.88     0    68  910 105.38 

-  .1088  -  0.155  +  0.058     19  237.101 

+  .1663  +  2.38  +  0.02  333 


3  459.454 

100  21  41.0 
23  27  31.68 
99  48  18.74 


68  910  104.50 
19  237.004 


-  .4029     -  0.81    +  0.05       83 

-  .0507  J  +  0.18    -  0.01         1 
+  . 1135  N+ 6.56    +1.34       48 


17 
9 

51 
23 


52.47 
16.92 
2.28 
53.02 


333 

83 

1 

48 


17 
9 

51 
24 


54.87 

16.16 

2.45 

0.92 


Definitive  values  of  the  secular  variations. 

89.  The  definitive  values  of  the  secular  variations,  as  inferred 
from  the  adopted  theories  and  the  concluded  values  of  the 
masses,  are  shown  in  the  following  table,  which  gives  in  detail 
the  parts  of  which  each  quantity  is  made  up. 

The  first  four  lines  of  the  table  give  the  values  of  the  secular 
variations  as  they  result  from  the  investigations  found  in  Vol. 
V,  Part  IV,  of  the  Astronomical  Papers,  after  correcting  the 
mass  of  each  planet  by  its  appropriate  factor. 

The  motion  of  the  perihelion  first  given,  denoted  by  Dt  n\, 
is  measured  along  the  plane  of  the  orbit  itself.  The  numbers 


89 1  SECULAR   VARIATIONS.  183 

given  being  divided  by  the  corresponding  values  of  the  eccen- 
tricity we  have  the  motion  of  the  perihelion  itself  along  the 
plane.  The  symbols  i0  and  #0  represent  the  inclinations  and 
longitudes  of  the  nodes  referred  at  each  epoch  to  the  ecliptic 
and  equinox  of  1850,  regarded  as  fixed.  The  motions  of  these 
elements  are  next  to  be  referred  to  the  fixed  ecliptic  of  the 
date.  So  referred,  they  are  designated  as  D?  i  and  D?  6.  The 
transformations  to  the  latter  quantities  are  made  by  comput- 
ing an  approximate  value  of  the  motion  of  the  node  due  to 
the  motion  of  the  ecliptic  alone  along  the  plane  of  the  orbit 
regarded  as  fixed. 
If  we  put 

«,,  the  inclination  of  the  fixed  orbit  of  the  planet  at  any  epoch 
TO  to  the  moving  ecliptic  at  any  time; 

61,  the  longitude  of  the  corresponding  node,  £h; 

F,  the  distance  from  the  node  Q  t  to  the  instantaneous  rota- 
tion axis  of  the  orbit  at  the  epoch  T0 ; 

we  shall  have 

Dt  v  =  H"  cosec  i\  sin  (L"  —  #1)  (a] 

If  we  compute  v0  and  H  from  the  equations 

H  sin  VQ  =  sin  i0  D?  00 
H  cos  r0  =  D?  i0 

and  then  find  Av  by  integrating  the  value  (a)  of  Dtr  from  1850 
to  the  date  we  shall  have 

sin  i  D?  #  =          H  sin  (  VQ  +  A  v] 
D°  i  =         H  cos  (vQ  -f  Av) 

The  change  of  Dt^  between  1850  and  the  extreme  epochs  has 
been  found  so  nearly  uniform  that  it  was  sufficient  to  multiply 
its  value  at  the  mid-epoch  (1675  or  1975)  by  2.5  to  obtain  Av. 

Next,  we  have  the  changes  in  i  and  0  due  to  the  motion  of  the 
ecliptic,  represented  by  T>]i  and  Df0,  and  computed  by  the 
formula 

Dlt  i=  —  Hf/GOS(r/f  -0) 
sin  i  D[  6  =  —  H"  cos  i  sin  (v"  —  0) 


184  DEFINITIVE  QUANTITIES.  [89 

The  planetary  precession  due  to  the  motion  of  the  ecliptic  is 
here  omitted,  to  be  afterwards  included  in  the  general  preces- 
sion. The  sum  of  the  two  motions  gives  the  actual  variation 
at  each  epoch,  referred  to  a  fixed  equinox. 

The  motion  of  6  itself  thus  found  is  increased  by  the  general 
precession,  which  gives  the  motion  of  6  at  each  epoch. 

The  motion  of  the  perihelion  to  be  actually  used  in  the  tables 
is  equal  to  the  motion  of  the  node  from  the  mean  equinox,  plus 
the  increase  of  the  arc  of  the  orbit  between  the  node  and 
perihelion.  The  adopted  value  of  this  quantity  is  found  by 
increasing  the  motion  of  n\  by  the  following  quantities: 

1.  The  change  due  to  the  motion  of  the  plane  of  the  orbit. 

2.  The  change  due  to  the  motion  of  the  ecliptic. 
The  formulae  for  these  two  quantities  are 

(1) ;  d]  Dt  n  =       tan  J  i$m  i  D?  d 

(2)  5  <?2  Dt  n  =  H"  tan  J  i  sin  (L"  -  0) 

3.  The  excess  of  motion  shown  by  observations  in  the  case 
of  Mercury  and  Mars,  and  computed  for  all  four  planets  as  if 
they  gravitated  toward  the  Sun  with  a  force  proportional  to 
r~n  where 

n  =  2.000  000  16120 

The  values  of  this  correction  are 

// 

Mercury;  Dt  n  —  43.37 
Venus;  16.98 

Earth;  10.45 

Mars;  5.55 

4.  The  general  precession. 

5.  In  the  case  of  the  Earth,  the  motion  arising  from  the 
action  of  the  Moon,  of  which  the  amount  is 

Dt  n"  =  7".68 

But  the  first  two  corrections  drop  out  in  this  case. 

The  preceding  transformations  of  the  secular  variations  are 
made  with  the  original  values  of  the  elements  e  and  i,  as  given 
in  Astronomical  Papers,  Vol.  V,  Part  IV,  pp.  337,  338. 


89J 


SECULAR  VARIATIONS. 


185 


Secular  variations  of  the  elements  of  the  four  orbits  at  the  three 
epochs,  1600,  1850,  and  3100,  as  inferred  from  the  definitively 
adopted  masses. 


Mercury. 


1600. 


1850. 


2100. 


Dte 

+       4.257 

+       4.227 

+       4.196 

tfDtTTi 

+  109.524 

+  109.498 

+  109.475 

DJtp 

21.581 

-     21.568 

-     21.551 

sinioD?0o 

54.891 

-     54.969 

55.049 

D?i 

-     21.786 

-     21.568 

-     21.347 

sintDfd 

—     54.813 

-     54.969 

-     55.130 

D{i 

+     28.884 

+     28.333 

+     27.785 

sin  t  DJ  B 

-     37.196 

-     37.397 

—     37.595 

Dt* 

+       7.098 

+       6.765 

+       6.438 

sin  i  Dt  0 

-     92.009 

-     92.366 

-     92.725 

JDt7T 

1.06 

1.06 

-       1.06 

Dt7T 

5593.41 

5598.70 

5604.02 

Dt0 

4262.98 

4266.12 

4269.24 

Venus. 

Dte 

-       9.959 

-       9.866 

-       9.772 

eDt7Ti 

-f       0.384 

+       0.219 

+       0.060 

Dfto 

-       2.484 

-       3.071 

-       3.656 

sin  to  DJ  00 

-     59.005 

-     59.112 

-     59.229 

D?t 

-       3.049 

-       3.071 

-       3.091 

sin*D?0 

-     58.978 

-     59.112 

-     59.260 

D<* 

+       6.690 

+       6.695 

+       6.697 

sin  i  DJ  0 

-     46.758 

-     46.582 

-     46.413 

Dti 

+       3.641 

+       3.624 

+       3.606 

sin  i  Dt  0 

-  105.736 

-  105.694 

-  105.673 

JDt7T 

-       0.36 

-       0.37 

-       0.38 

BtTT 

5090.07 

5072.44 

5054.92 

Dt0 

3230.39 

3237.98 

3245.22 

186  DEFINITIVE  RESULTS.  |89,  90 

Secular  variations  of  the  elements  of  the  four  orbits,  etc.— Cont'd. 

Earth. 
1600.  1850.  2100. 


Dte" 

-   8.467 

-   8.595 

-   8.727 

e"Dt7r" 

Dt7T" 

+  19.293 
6179.58 

+  19.210 
6187.41 

+  19.139 
6195.68 

H"  sin  Li' 
H"  cos  Li' 

+   4.370 
47.113 

+   5.341 
-  46.838 

+   6.305 
—  .  46.550 

log  H" 

L'o 
L" 

1.67500 
1740  42'.04 
171°  12/.83 

1.67340 
173°  29'.68 
1730  29X.68 

1.67187 
1720  17M8 
1750  46'.62 

Po 

p 

5034.91 

5018.28 
-  46.761 

5036.13 
5023.82 
-  46.838 

5037.36 
5029.38 
-  46.847 

Mars. 

e  Dt  7T] 

+  18.775 
+•  148.633 
28.994 

+  18.706 
+  148.707 
-  29.396 

+  18.623 
+  148.762 
-  29.803 

sin  *o  D?  00 

34.023 

34.012 

34.017 

D?* 

-  29.482 

-  29.396 

-  29.309 

sin  t  D?  0 

-  33.605 

34.012 

34.445 

DM 
sin  i  DJ  0 

+  26.964 

-  38.860 

+  27.104 
38.551 

+  27.245 
38.247 

Dtt 

-   2.518 

-   2.292 

2.064 

sin  i  Dt  0 

72.465 

-  72.563 

-  72.692 

1>°^ 

+   0.08 
6621.51 

+   0.07 
6623.96 

+   0.06 
6626.25 

Dt  0 

2776.39 

2776.87 

2776.63 

Secular  acceleration  of  the  mean  motions. 

90.  The  mean  motions  of  the  planets,  like  that  of  the  Moon,  are 
subject  to  a  secular  acceleration  arising  from  the  secular  vari- 
ations of  the  elements  of  the  orbits.  The  following  formulae 
for  this  acceleration  are  formed  by  differentiating  the  known 


90]  SECULAR   ACCELERATIONS.  187 

expressions  for  the  variation  of  the  longitude  of  the  epoch  in 
the  theory  of  the  variation  of  elements.    The  notation  is  that 
of  Astronomical  Papers,  Vol.  V,  Part  IV. 
We  compute  for  the  action  of  an  outer  on  an  inner  planet: 

A  =  D  <*\} 

B  =  -  (D  -  D2  —  2  D3)  c(°> 

8 

W-  -  (2  -  9  D  +  3  D2  +  4  D3)  c^ 

8 

Then 

D;  «„  =  w'  a  n  Dt{  A  <J2  +  Be2  -  Ge'2  +  Wee'  cos  (n  -  n')\ 
For  the  action  of  an  inner  on  an  outer  planet  we  compute 

A'  =-^(l  +  D)6^) 

B7  =      l  (D  +  2  D2  +  D3)  c(»] 
4 

8 

W  =      I  (10  +  3  D  -  9  D2  -  4  D3)  <t\} 
8 

D?  10  =  m  n'  Dt  j  A7  a2  +  B'e2  +  We'2  +  Wee'  cos  (n  -  *')  [ 

The  symbol  Dt  indicates  the  secular  variation  of  the  expres- 
sion following  it  produced  by  the  action  of  all  the  planets.  The 
unit  of  time  must  be  the  same  one  in  which  n  is  expressed. 

The  following  table  gives  the  results  of  this  computation : 

Secular  change  of  the  centennial  mean  motions. 
Action  of—     Mercury.  Venus.  Earth.  Mars. 


Venus,      —0.0426 

. 

-0.0104 

+  0.0010 

Earth,      -0.0029 

+  0.0128 

.     .     . 

+  0.0119 

Mars,        +  0.0003 

-0.0001 

-  0.0012 

. 

Jupiter,    -0.0039 

—  0.0046 

-0.0308 

+  0.0004 

Saturn,     -0.0004 

+  0.0015 

+  0.0021 

+  0.0036 

Total,        -0.0495         +0.0096         -0.0403         +0.0169 


188  DEFINITIVE   QUANTITIES.  [91,  92 

The  measure  of  time. 

91.  The  fictitious  mean  Sun  whose  transit  over  any  meridian 
defines  the  moment  of  mean  noon  on  that  meridian  is  a  point 
on  the  celestial  sphere  having  a  uniform  sidereal  motion  in  the 
plane  of  the  Earth's  equator,  and  a  Eight  Ascension  as  nearly 
as  may  be  equal  to  the  Sun's  mean  longitude.     If  we  put  /*  for 
this  uniform  sidereal  motion  and  add  to  JA  the  precession  of  the 
equinox  in  Eight  Ascension  we  have  for  the  mean  Eight  Ascen- 
sion of  this  fictitious  mean  Sun 

T  =  TO  +  /i  T  +  4606//.36  T  +  1".394  T2 

From  §§  88,  90,  and  100  the  expression  for  the  Sun's  mean 
longitude,  affected  by  aberration,  is  found  to  be 

L  =  2790  47'  58".2  +  129602766".74  T  +  1//.089  T2 

Equalizing  the  coefficients  of  T  we  find,  for  the  mean  Eight 
Ascension  of  the  fictitious  mean  Sun 

r  =  279°  47'  58".2  +  1296027G6//.74  T  +  l//.394  T2 

This  differs  from  the  mean  longitude  of  the  actual  Sun  by  the 
quantity 

r  -  L  =  0".305  T2  =  08.020  T2 

This  difference  is  of  no  importance  in  the  astronomy  of  our 
time,  but  may  result  in  an  error  of  2s  in  the  course  of  one  thou- 
sand years  in  the  measurement  of  time  by  the  actual  mean 
sun.  We  must  leave  to  the  astronomers  of  the  future  the 
question  how  best  to  meet  the  question  thus  arising.  Chang- 
ing to  time  the  expression  for  r,  the  difference  or  mean  excess 
of  sidereal  over  mean  time  for  the  meridian  of  Greenwich 
becomes 

T  =  l&  39ra  118.880  +  24"  Om  K84449  t  +  08.0929  T2 

t  being  time  in  Julian  years  after  1850,  January  0,  Greenwich 
mean  noon. 

Constant  of  aberration. 

92.  We  first  investigate  certain  fundamental  constants  con- 
nected with  the  motion  of  the  Sun,  Earth,  and  Moon,  on  which 
the  precession  and  nutation  depend. 


92,  93]  MASS   OF   THE  MOON.  189" 

From  the  adopted  value  of  the  solar  parallax, 

n  =  8//.790, 
and  the  adopted  velocity  of  light  in  kilometers  per  second, 

Y  =  299  860, 

follows  for  the  constant  of  aberration  the  value 

A  =  20//.501 

But  if  we  accept  the  mean  result  of  the  solutions  of  §  83  as 
giving  the  most  likely  value  of  the  solar  parallax,  we  shall 
have 

n  =  8".7854 

Then  §  75  will  give 

A  =  20".511 

as  the  adjusted  value  of  the  constant  of  aberration. 
Mass  of  the  Moon. 

93.  By  means  of  the  equation  of  §  71  between  the  lunar 
inequality  P  in  the  motion  of  the  Earth  and  the  mass  of  the 
Moon 

/*'P  =  [1.78207]  n 

we  may  find  a  fresh  value  of  the  Moon's  mass  from  the  values 
of  7t  and  P. 
We  have  found  from  observation 

P  =  6".465  i  .015 

Thus  follows,  for  the  mass  of  the  Moon,  when  7r  =  8".790, 
yw  =  1  :  81.32  i  0.20 

Combining  this  with  the  value  found  from  the  constant  of 
nutation, 

yw  =  1  :  81.58  ±  0.20 

we  have,  as  the  definitive  mass  of  the  Moon, 
*  =  !:  81.45  i  0.1 


190  DEFINITIVE   QUANTITIES.  [94,95 

Parallactic  inequality  of  the  Moon. 

94.  From  the  transformation  of  HANSEN'S  lunar  theory  in 
Astronomical  Papers,  Vol.  I,  it  may  be  concluded  that  the  solar 
parallax  and  the  parailactic  inequality  are  connected  by  the 
relation 

P.  I.  =  [1.16242]  ln£  7t 

1  +  V 

=  [1.15176]  7i 

Hence  we  have,  for  the  coefficient  of  the  parailactic  inequality 
of  the  Moon,  corresponding  to  n  =  8//.790, 

124".66 

Here  the  inequality  is  that  in  ecliptic  longitude. 
The  centimeter -second  system  of  units. 

95.  There  are  certain  methods  in  physics  by  which  the  next 
step  in  the  course  of  our  researches  will  be  guided.    The  adop- 
tion of  a  system  of  absolute  units  has  simplified  the  methods 
and  conceptions  of  physics  to  such  an  extent  that  we  may 
find  it  advantageous  to  introduce  a  similar  system  into  those 
investigations  of  astronomy  which  are  closely  connected  with 
that  science. 

The  fundamental  units  most  widely  adopted  are  the  centi- 
meter as  the  unit  of  length,  the  gram  as  the  unit  of  mass, 
and  the  second  as  the  unit  of  time.  There  is,  however,  an 
insuperable  difficulty  in  the  way  of  introducing  the  gram, 
or  any  other  arbitrary  terrestrial  unit  of  mass,  into  astronomy, 
from  the  fact  that  the  astronomical  masses  with  which  we  are 
concerned  can  not  be  determined  with  sufficient  precision  in 
units  of  terrestrial  mass.  It  is,  therefore,  quite  common  in 
celestial  mechanics  to  regard  the  unit  of  mass  as  arbitrary, 
and  to  multiply  this  arbitrary  unit  by  a  factor  which  will 
represent  its  attractive  force  upon  a  unit  particle  at  unit  dis- 
tance. The  introduction  of  this  factor  is,  however,  needless. 
It  is  simpler  to  adopt  the  course  of  DELAUNAY  and  many  other 
writers,  and  regard  the  unit  of  mass  as  a  derived  one,  based 
on  the  units  of  time  and  length,  by  defining  it  as  that  mass 
which  will  attract  an  equal  mass  at  unit  distance  with  force 


95,  96]  MASSES  OF  THE  EARTH  AND  MOON.  191 

unity.  In  this  definition  the  unit  of  force  retains  its  physical 
meaning,  as  that  force  which,  acting  on  unit  mass,  will  pro- 
duce a  unit  of  acceleration  in  a  unit  of  time. 

The  number  of  fundamental  units  is  then  reduced  to  two, 
those  of  time  and  length,  and  the  unit  mass  becomes  a:derived 
one  of  dimensions, 


The  centimeter  as  a  unit  of  length  wou 
small  for  astronomical  purposes,  if  we  had  to  deal  mainly  rwith 
natural  numbers,  but  it  causes  no  inconvenience  in  logarith- 
mic computations,  and  has  the  advantage  of  being  assimilated 
directly  to  the  centimeter-gram-second  system  in  physics. 
We  shall  therefore  adopt  it,  expressing  our  results,  however. 
in  terms  of  other  units  whenever  convenience  will  thereby  be 
gained. 

I  shall  make  clear  this  assimilation  and  the  use  of  the  unit 
of  mass  as  a  derived  one,  by  calling  this  the  centimeter- 
second  system. 

In  the  latter  the  definitions  of  units  in  the  centimeter- 
gram-second  system  will  remain  unchanged,  except  that  the 
derived  unit  of  mass  must  be  substituted  for  the  gram.  The 
dimensions  of  units  in  the  centimeter-second  system  will  be 
found  by  making  the  above  substitution  for  M  in  the  expres- 
sions for  those  of  the  centimeter-  gram  -second  system. 

Masses  of  the  Earth  and  Moon  in  centimeter  -second  units. 

96.  A  fundamental  quantity  in  the  centimeter-  second  system 
is  the  mass  of  the  Earth.  This  mass  will  be  by  definition  the 
force  of  gravity  of  the  Earth,  if  concentrated  in  a  point  at  the 
distance  of  one  centimeter.  Were  the  Earth  a  sphere  of  known 
dimensions,  it  could  be  readily  determined  through  the  force 
of  gravity  at  any  point  on  its  surface.  This  being  not  the  case, 
we  shall  proceed  on  the  accepted  approximate  theory  that  the 
geoid  is  an  ellipsoid  of  revolution,  and  that  the  force  of  gravity 
at  a  point  the  sine  of  whose  latitude  is  1  :  -v/3  is  the  same  as 
if  the  mass  of  the  Earth  were  concentrated  in  its  center. 

The  determination  of  this  constant  with  astronomical  preci- 
sion is  a  difficult  and  we  might  say  hitherto  an  insoluble  prob- 


192  DEFINITIVE  Q  UANTITIES.  [96 

lem,  owing  to  the  heterogeneity  of  the  Earth  and  the  absence 
of  determinations  of  the  force  of  gravity  over  the  surface  of  the 
ocean.  Although  the  limits  of  uncertainty  thus  arising  can 
not  be  set  with  any  approach  to  precision,  I  do  not  think  they 
are  such  as  to  greatly  impair  the  astronomical  results  which 
are  to  be  derived  from  them.  Investigations  in  geodesy  not 
being  practicable  in  the  present  work,  I  have,  mainly  from  a 
study  of  the  work  of  G.  W.  HILL,*  assumed  for  the  length  of 
the  seconds  pendulum  at  the  point  the  sine  of  whose  latitude 
is  1  :  V3,  which  I  shall  call  the  mean  latitude, 

L!  =  99.2715 

With  this  we  may  compare  HELMERT'S  expression  for  the 
length  of  the  seconds  pendulum  in  terms  of  the  latitude 

L  =  Om.990918  (1  +  .005310  sin  2tp) 
which  gives 

L!  =  99.2688 

From  these  values  of  LI  we  have: 

HILL.  HELMERT. 

Gravity  at  mean  latitude,               979.770  979.745 

Correction  for  centrifugal  force,          2.260  2.260 

Attraction  of  the  Earth,                 982.030  982.005 

I  also  accept  as  the  result  of  CLARKE'S  investigation  of  1880, 

Equatorial  radius  of  the  Earth,      6378249m 
Reduction  to  mean  latitude,  7245 

Mean  radius  of  the  Earth,  6371004 

From  HILL'S  and  HELMERT'S  numbers  follows : 

Logarithm  mass  of  Earth  expressed  in  centimeter- second  units. 

HILL.  HELMERT. 

20.600541.  20.600530. 

*  Astronomical  Papers,  Vol.  Ill,  p.  339. 


96,  97  PARALLAX  OF  THE  MOON.  193 

From  the  adopted  ratio  of  the  mass  of  the  Moon  to  that  of  the 
Earth: 

fji  =  1 :  81.45 

follows 
Logarithm  of  the  mass  of  the  Moon  in  centimeter-second  unite, 

18.68965. 
Parallax  of  the  Moon. 

97.  From  these  results  the  distance  of  the  Moon  and  the 
relation  between  the  mass  and  distance  of  the  sun  follow  in  a 
very  simple  way.  By  the  formulae  of  elliptic  motion  it  follows 
that  when  we  put 

w,w',  the  masses  of  any  two  bodies  revolving  around  each 

other  in  virtue  of  their  mutual  gravitation ; 
a,  the  sernimajor  axis  of  the  relative  orbit,  which  would 

be  the  actual  distance  if  the  motion  were  circular; 
n,  their  mean  angular  motion  in  unit  of  time; 

we  have  the  relation 

a3  ril  =  m  +  m' 

This  relation  is  rigorous  and  independent  of  the  adopted  units 
of  length  and  time,  provided  we  define  the  unit  of  mass  in  the 
way  already  done.  It  follows  that  if  the  Moon  in  its  revolu- 
tion around  the  Earth  were  not  subject  to  disturbance,  its  mean 
motion  in  one  second,  and  its  distance  expressed  in  centimeters, 
would  be  connected  by  the  relation 

Log  a3  n2  =  log  m"  ( 1  +  /*)  =  20.605841 

In  the  theories  of  DELAUNAY  and  ADAMS  the  quantity  &,  as 
determined  by  this  equation,  is  accepted  as  a  fundamental 
element,  and  it  is  shown  that  in  consequence  of  the  perturba- 
tions produced  by  the  Sun  the  constant  770  of  the  Moon's  hor- 
izontal parallax  is  connected  with  a  by  the  relation 

a  sin  770  =  1. 000907  p 

p  being  the  radius  of  the  Earth  corresponding  to  770 
5690  N  ALM 13 


194  DEFINITIVE   QUANTITIES.  [97,  98 

From  the  mean  sidereal  motion  of  the  Moon  in  a  Julian 
century 

1336  .  85136  revolutions 

we  find,  for  the  co-logarithm  of  the  motion  in  arc  in  one 
second 

log  JL=  5.574841 

and  thus  have  for  the  undisturbed  mean  distance  of  the  Moon 
in  centimeters 

log  a  =  10.585174 

and  hence 

log  sin  770  =  8.219921 

/    // 

770  =    57  2.68 

Red.  to  sine,  —  .16 

Constant  of  sin  n  in  arc,        57  2.52 

Using    HELMERT'S   length  of  the  seconds  pendulum  we 
should  have  found  for  this  constant 

3422".55 
Mass  and  parallax  of  the  Sun. 

98.  In  the  case  of  the  motion  of  the  center  of  gravity  of  the 
Earth  and  Moon  around  the  Sun  the  relation  of  §97  becomes 

a'3  n12  =  M]  -f  m"  (1  +  ^) 

MI  being  the  mass  of  the  Sun.  Replacing  a'  by  TT,  the  parallax 
of  the  Sun,  and  p  the  radius  of  the  Earth,  we  find  for  the 
ratio  M  of  the  mass  of  the  Sun  to  the  sum  of  the  masses  of  the 
Earth  and  Moon 


M 

m"  (I  4-  IJL)  sin3  n  " 

log  M7r3  =  8.349674 


98, 99]  SUN'S  MASS  AND  PARALLAX.  195 

The  values  of  M  corresponding  to  certain  values  of  the  mean 
equatorial  horizontal  parallax  of  the  Sun  are  as  follows : 


M 


8.780 

330514 

8.785 

329951 

8.790 

329388 

8.795 

328827 

8.800 

328266 

Nutation  and  mechanical  ellipticity  of  the  Earth. 

99.  Begarding  the  mass  of  the  Moon  as  known,  we  now 
utilize  the  equations  of  §  67  to  obtain  the  constant  of  nutation 
and  the  mechanical  ellipticity  of  the  Earth.  The  last  two  of 
these  equations  give,  for  the  absolute  precessional  constant, 
when  the  Julian  year  is  the  unit  of  time, 

p  =  [[5.975052]  j-^—  +  5310".o]  °~A 

* 
We  have  found,  in  §  66,  for  a  Julian  year 

p  =  54".8990 
We  then  have,  for  the  mechanical  ellipticity  of  the  Earth, 

0  ~  A  =  0.0032753 


We  also  have,  from  the  first  equation  of  §  66,  for  the  constant 
of  nutation  for  1850 

N  =  9".214 

For  the  parts  of  the  precessional  constant  which  arise  from 
the  action  of  the  Sun  and  of  the  Moon,  respectively,  we  have — 

// 

Action  of  the  Sun 17.3919 

Action  of  the  Moon  .  37.5071 


196 


DEFINITIVE   QUANTITIES. 

Precession. 


[100, 101 


100.  In  order  to  develop  the  terms  of  the  precession  and 
obliquity  to  higher  powers  of  the  time,  I  have  extended  their 
computation  one  step  backward  and  forward  from  the  three 
fundamental  epochs,  by  extrapolation  of  H  and  L.  The  results 
are  as  follows  : 


Year. 


Motion  of  the  ecliptic  and  equator. 


log.  K 


1350 

1.67666 

168  56.13 

-  46.613 

2009.05 

1600 

1.67500 

171  12.84 

-  46.761 

2006.92 

1850 

1.67340 

173  29.68 

-  46.838 

2004.79 

2100 

1.67187 

175  46.63 

-  46.847 

2002.66 

2350 

1.67039 

178  3.50 

-  46.789 

2000.52 

Centennial  precessions  for  tropical  centuries. 


Year. 

1350 
1600 
1850 
2100 
2350 


In  longitude— 

Lunisolar. 

Planetary. 

General. 

5033.58  * 

-  20.94 

5012.64 

5034.80 

-  16.63 

5018.17 

5036.02 

-  12.31 

5023.71 

5037.25 

-    7.98 

5029.27 

5038.49 

-    3.67 

5034.82 

In  Right 
Ascension. 

4592.41 
4599.38 
4606.36 
4613.35 
4620.32 


From  these  values  we  have  the  following  general  expres- 
sions : 


Annual  precession  in  Eight  Ascension; 
Annual  precession  in  longitude; 
Centennial  precession  in  longitude; 
Total  precession  from  1850; 


46.0636  +  0.0279  T 
50.2371  +  0.0222  T 
5023.71      +2.218   T 
5023.71  T  +  1.109   T2 


Mean  obliquity  of  the  ecliptic. 

101.  The  expression  for  the  mean  obliquity  when  T  is  counted 
from  1900  is— 


e  =  23°  27'  8".26  -  46".845T  -  0".0059  T2  +  0".00181  T3 


101,  102]  PRECESSION.  197 

Tables  of  the  mean  obliquity  at  different  epochs. 


Year. 

Obliquity. 

Year. 

Obliquity. 

0    /       // 

o    /     // 

1600 

23  29  28.69 

-2500 

23  58  44.00 

1650 

29  5.31 

-2000 

55  38.99 

1700 

28  41.91 

-  J500 

52  23.10 

1750 

28  18.51 

-1000 

48  57.70 

1800 

27  55.10 

-  500 

45  24.14 

1850 

27  31.68 

0 

41  43.78 

1900 

27  8.26 

+  500 

37  57.97 

1950 

26  44.84 

1000 

34  8.07 

2000 

26  21.41 

1500 

30  15.43 

2050 

25  57.98 

2000 

26  21.41 

2100 

23  25  34.56 

2500 

23  22  27.37 

Relative  positions  of  the  equator  and  ecliptic  at  different  dates. 

102.  The  motions  expressed  iii  the  preceding  tables  are,  for 
the  most  part,  purely  instantaneous  ones,  referred  to  the  planes 
of  the  ecliptic  and  equator  of  each  separate  epoch.  For  the 
reduction  of  the  places  of  the  fixed  stars  from  one  epoch  to 
another,  it  is  necessary  to  know  the  relative  position  of  the 
planes  of  the  equator  or  ecliptic  at  the  two  epochs.  We  shall 
therefore  derive  the  fundamental  quantities  which  express 
the  position  of  the  equator  and  the  ecliptic  at  any  one  epoch 
relatively  to  their  positions  at  a  fundamental  epoch  taken  at 
pleasure.  The  latter  we  shall  call  zero  position.  Then,  the 
zero  equator  and  ecliptic  are  those  of  the  fundamental  epoch; 
the  equator  and  ecliptic  simply  those  of  any  other  varying 
epoch.  So  far  as  convenient,  and  as  conducive  to  ease  in 
comparing  our  results  with  former  ones,  we  shall  use  the  nota- 
tion of  BESSEL. 

To  derive  the  equations  for  the  motions,  let  us  consider  the 
following  four  points  of  the  celestial  sphere: 

E0,  the  pole  of  the  zero  ecliptic. 

E,  the  pole  of  the  actual  ecliptic. 

P0,  the  pole  of  the  zero  equator. 

P,  the  pole  of  the  actual  equator. 


198  DEFINITIVE   QUANTITIES.  |102 

We  put, 

fT  =  PE0,  the  obliquity  of  the  equator  to  the  zero  ecliptic; 
k  =  EE0,  the  inclination  of  the  two  ecliptics; 

770,  the  longitude  of   the  node  of  the  ecliptic  on  the  zero 

ecliptic,  measured  from  the  zero  equinox  of  the  date; 

771,  the  longitude  of  the  same  node,  measured  from  the  actual 

equinox; 

A,  the  arc  of  the  equator  intercepted  between  the  two  eclip- 
tics, or  the  planetary  precession  on  the  equator ; 

if?a  the  total,  lunisolar  precession  on  the  zero  ecliptic  from 
the  zero  epoch  to  the  actual  epoch ; 

w,  the  rate  of  motion  of  the  pole  of  the  equator; 

T,  the  time,  expressed  in  units  of  250  years  from  the  zero 
epoch  to  any  other  epoch. 

The  position  of  the  variable  point  E  is  denned  by  the  quan- 
tities k  and  J70  or  77i,  which  are  themselves  to  be  determined 
through  the  values  of  x  and  L  of  §  100. 

The  position  of  the  variable  point  P  is  determined  by  the 
condition  that  its  motion  is  constantly  at  right  angles  to  the 
arc  EP,  and  its  velocity  measured  on  the  arc  of  a  great  circle 
is  given  by  the  equation 

ds 

=  n  =  P  sin  s  cos  s  (a) 

ci  t 

The  positions  of  the  equator  and  equinox  relative  to  the 
zero  equator  and  ecliptic  are  then  determined  by  the  quanti- 
ties £1,  ij>  and  A.  The  spherical  triangle  P  E0  E  gives  the  follow- 
ing equations: 

sin  A,        sin  77!        sin  7I0 

sin  k  "      sin  e i          sin  e 

During  a  period  of  several  centuries  the  quantities  k  and  A  are 
so  small  that  no  distinction  is  necessary  between  them  and 
their  sines.  We  may  therefore  put 

A  =  k  sin  77t  cosec  £1  =  k  sin  770  cosec  e  (b) 

We  also  have,  from  the  law  of  motion  of  the  pole  of  the 

equator, 

Dt  £t  =  n  sin  A 

Dt  fy  =  n  cos  A  cosec  e\ 


102] 


MOTION   OF    THE   EQUATOR. 


199 


As  the  value  of  81  does  not  change  by  0".6  from  one  epoch  to 
another,  we  may,  without  appreciable  error,  use  f0  for  ^  in  the 
formulae  (b)  and  (c).  To  use  these  equations,  we  first  obtain  k 
and  77!  from  the  secular  motion  of  the  ecliptic,  while  n  is  com- 
puted for  any  epoch  from  the  formula  (a).  We  then  easily 
develop  the  values  of  s-i  and  ip  in  powers  of  the  time  by  the 
equations  (c).  The  values  of  n  have  no  reference  to  any 
special  coordinates.  From  the  table  ot  §  100  it  will  be  seen  that 
we  may  put 

n  =  2004".79  -  2".13  r' 

r'  being  counted  from  1850. 

To  find  the  value  of  III  in  each  case,  we  remark  that  the 
instantaneous  values  of  L  given  in  §  100  show  that  the  instan- 
taneous node,  or  intersections  of  two  consecutive  ecliptics, 
moves  with  so  near  an  approach  to  uniformity  that  we  may 
take  for  the  actual  node  between  the  ecliptics  of  any  two 
epochs  TI  and  r2  the  mean  of  the  instantaneous  nodes  for  those 
two  epochs.  For  example,  let  it  be  required  to  find  the  value 
of  77j  for  the  node  of  the  ecliptic  of  2100  on  that  of  1850.  We 
have 


For  2100 

For  1850,  referred  to  eq.  of  2100 
Concluded  value  of  77i       ... 


L  =  175  46.63 
L  =  176  59.13 
77!  =  176  22.9 


As  the  basis  of  our  work  we  have  computed  the  required 
quantities  for  the  zero  ecliptics  of  1600,  1850,  and  2100, 
respectively.  The  values  of  k  and  77i  for  the  ecliptics  of  two 
hundred  and  fifty  years  before  and  after  these  epochs  are  as 
follows : 


Zero  epoch. 


1600 
1850 

2IOO 


—  250  Y 

+  25OY 

k 

n, 

k 

n, 

ff 
-118.48 
—  118.07 
-117.64 

0                 / 

1  68  20.0 
170  36.7 
172  53-4 

// 
+  118.07 
+  117.64 

+  117-23 

0                / 

174    5-9 

176    22.  9 
178    39.9 

200  DEFINITIVE   QUANTITIES.  [102 

Changing  the.  unit  of  time  to  two  hundred  and  fifty  years, 
the  equations  (a)  (b)  and  (c)  give  the  following  values  of  the 
derivatives  of  fi  and  : 


Zero-epoch.         —  250  Y  +250Y  —  250  Y  +250Y 

1600  _  1.4636  +  0.7400  12600.33  12573.65 
1850  -1.1768  +0.4527  12603.44  12576.65 
2100  -0.8898  +0.1665  12606.57  12579.71 

At  the  respective  epochs  T>T£\  vanishes,   and  Dr#  has  the 
values  of  the  luuisolar  precession  in  longitude  (§  100). 
Developing  in  powers  of  r  we  have- the  following  results: 

Zero-epoch.          o      /        //  //  // 

1600;   el  =  23  29  28.69-+  0.5509  r2  -  0.1206  r3 
1850;   ei  =  23  27  31.68  +  0.4074      -  0.1207 
2100;   €t  =  23  25  34.56  +  0.2641      -  0.1206 

1600;    #  =  12587.00  T  -  6.67  r2 
1850;    #  =  12590.05     -6.70 
2100;   #  =  12593.14     -6.72 

//  // 

1600;    A  =  45.28  r  -  14.83  T* 
1850;    A  =  33.52     -  14.86 
2100;    A  =  21.75     -14.88 

These  values  of  Si  and  #  completely  fix  the  position  of  the 
equator  at  the  time  T  relative  to  the  zero  ecliptic  and  equinox. 
For  the  reduction  of  coordinates  from  one  epoch  to  another 
we  must  express  the  position  of  the  equator  at  the  time  r.  We 
consider  the  triangle  PE0P0,  of  which  the  sides  and  opposite 
angles  are  designated 

Sides,  £0  fi  0 

Opposite  angles,     90°  -  C        90°  —  £,        # 

If,  in  the  Gaussian  relations  between  the  parts  of  this  triangle, 
we  put 

sin  A  (f,  —  £Q}  =  A  (fi  —  e«)  =  A  z/£ 


102]  MOTION   OF   THE  EQUATOR.  201 

and  regard  the  cosine  of  this  angle  as  unity,  we  have 

tan  J  (C  +  Ci)  =  cos  J  (et  +  e0)  tan  £  ip 

If  we  develop  the  differences  between  the  tangent  and  the 
arc  we  find  from  these  equations 

r  +  c,  =  ^  cos  A  (fi  +  f0)  (1  - 


where  we  put  z0  for  the  approximate  value  of  C  —  Ci 

For  the  inclination  6  of  the  mean  equator  of  the  epoch  r  to 
the  zero  equator,  we  have  the  equation 


sin  9  = 

cos 


and  then,  by  developing  in  powers  of  6  and  ^,  we  find 


=  ip  sin  f0  (1  +  4  C2)  (1  —  i  ^2  cos2  f0) 
We  thus  find 

Zero-epoch.  //  //  // 

1600;  C  +  Ci  =  11543.79  T  -  6.12  r2  +  0.57  r3 
1850;  11549.44      -  6.L4      +0.57 

2100;  11555.12      -6.16      +0.58 

1600;  C  -  Ci  =        45.29  r  -  9.92  r2 
1850;  33.53     -9.93 

2100;  21.76     -  9.94 

1600;      e       =    5017.30  T  -  2.66  r2  -  0.64  r3 
1850;  5011.97      -  2.67       -  0.64 

21CO;  5006.64      -2.67       -0.65 

To  show  the  significance  of  the  preceding  quantities,  con 
sider  once  more  the  spherical  quadrangle  P0  E0  EP.    Let  these 


202  DEFINITIVE  QUANTITIES.  [102 

letters  represent  the  positions  of  the  poles  on  the  celestial 
sphere  at  any  two  epochs.    In  this  quadrangle  we  shall  have 

Angle  E0P0E   =  90°  -  Ci 

Angle  E  P  P0  =  90°  -  C  +  A 
SideP0P  =  # 

Let  S  be  the  position  of  a  star  on  the  celestial  sphere.  Its 
polar  distances  at  the  two  epochs  will  be  P0  S  and  P  S  and  its 
Eight  Ascensions  will  be  determined  by  the  angles  P0  and  P 
of  the  triangle  S  P0  P. 

Thus,  if  the  Eight  Ascension  and  Declination  of  S  are  given 
for  one  epoch,  we  can  find  it  for  the  other  epoch  by  the  solu- 
tion of  the  triangle  S  P  P0  when  we  have  given  the  values  of 
the  quantities  0,  Ci,  and  £  -f  A. 

To  find  the  values  of  these  quantities  from  the  preceding 
formula,  let  T  be  the  zero-epoch,  expressed  in  calendar  years, 
and  let  -c  be  the  interval  between  the  two  epochs,  taken  posi- 
tively when  the  zero-epoch  is  the  earlier  one,  and  negatively 
when  it  is  the  later  one.  We  interpolate  the  coefficients  of  r 
and  its  powers  from  the  preceding  formula  to  the  epoch  T. 
Then  by  substituting  the  value  of  r  in  the  formula  we  shall 
have  the  values  of  the  required  quantities,  and  hence  the  data 
for  reducing  the  position  of  S  from  one  epoch  to  the  other. 


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MAT 


REC'D  LD 

DEC  19  1957 


otf 


JUN     2 


8EC.CB.ltOf    5T8 


1948 


75m-7,'30 


U.C.BERKELEY  LIBRARIES 


COL13S63L7 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


